3. INNER PRODUCT SPACES


 Brittney Goodwin
 1 years ago
 Views:
Transcription
1 . INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space. In R and R we have the concepts of lengths and angles. In those spaces we use the dot product for this purpose, but the dot product only makes sense when we have components. In the absence of components we introduce something called an inner product to play the role of the dot product. We consider only vector spaces over C, or some subfield of C, such as R. An inner product space is a vector space V over C together with a function (called an inner product) that associates with every pair of vectors in V a complex number u v such that: () v u u v for all u, v V; () u v w u w v w for all u, v, w V; () u v u v for all u, v V and all C; (4) v v is real and for all v V; (5) v v = if and only if v =. These are known as the axioms for an inner product space (along with the usual vector space axioms). A Euclidean space is a vector space over R, where u v R for all u, v and where the above five axioms hold. In this case we can simplify the axioms slightly: () v u u v for all u, v V; () u v w u w v w for all u, v, w V; () u v u v for all u, v V and all C; (4) v v for all v V; (5) v v = if and only if v =. Example : Take V = R n as a vector space over R and define u v = u v u n v n where u = (u,, u n ) and v = (v,, v n ) (the usual dot product). This makes R n into a Euclidean space. When n = we can interpret this geometrically as the real Euclidean plane. When n = this is the usual Euclidean space. Example : Take V = C n as a vector space over C and define u = (u,, u n ) and v = (v,, v n ). u v u v... u v n n where Example : Take V = M n (R), the space of nn matrices over R where A B = trace(a T B). Note, this becomes the usual dot product if we consider an n n matrix as a vector with n components, since trace(a T B) = a b ij ij if A = (a ij ) and B = (b ij ). n i, j Example 4: Show that R can be made into a Euclidean space by defining u u = 5x x x y x y + 5y y when u = (x, y ) and u = (x, y ). Solution: We check the five axioms. 9
2 () u u = 5x x x y x y + 5y y = u u. () If u = (x, y ) then u + u u = 5(x + x )x x (y + y ) (x + x )y + 5(y + y )y = 5x x + 5x x x y x y x y x y + 5y y + 5y y = (5x x x y x y + 5y y ) + (5x x x y x y + 5y y ) = u u + u u. () u u = 5(x )x x (y ) (x )y + 5(y )y = [5x x x y x y + 5y y ] = u u. (4) If v = (x, y) then v v = 5x xy + 5y = 5(x xy/5 + y ) = 5(x y/5) + 4y /5 for all x, y. (5) v v = if and only if x = y/5 and y =, that is, if and only if v =. Now we move to a rather different sort of inner product, but one that still satisfies tha above axioms. Inner product spaces of this type are very important in mathematics. Example 5: Take V to be the space of continuous functions of a real variable and define u x v( x) NOTE: Axioms (), () show that the function u However u( x) v( x) dx u v is a linear transformation for a fixed v. u v u is not linear since v u u v u v u v v u... Lengths and Distances The length of a vector in an inner product space is defined by: (Remember that v v v. v v is real and nonnegative. The square root is the nonnegative one.) So the zero vector is the only one with zero length. All other vectors in an inner product space have positive length. Example 6: In R, with the dot product as inner product, the length of (x, y, z) is x + y + z. Example 6: If V is the space of continuous functions of a real variable and u x vx uxvx dx = f(x)g(x) dx then the length of f(x) = x is x 4 dx = 5. The following properties of length are easily proved. Theorem : For all vectors u, v and all scalars : () v =. v ; () v ; () v = if and only if v =.
3 Theorem (Cauchy Schwarz Inequality): u v u. v. Equality holds if and only if u = u v Proof: Let d = v. Now u dv = u dv u dv = u u d v u d u v + dd v v = u dd v + dd v = u d v = u u v v Since u dv, u v u v. Example 7: In R n we have y x y Example 8: f(x)g(x) dx x. i i f(x) dx g(x) dx. i i u v v v. The Triangle Inequality in the Euclidean plane states no side of a triangle can be longer than the sum of the other two sides. It is usually proved geometrically, or appealing to the principle that the shortest distance between two points is a straight line. In a general inner product space we must prove it from the axioms. Theorem (Triangle Inequality): For all vectors u, v: u + v u + v. Proof: u + v = u + v u + v = u u + v v + u v + v u = u + v + Re(u v) u + v + u v u + v + u. v ( u + v ) So u + v u + v. We define the distance between two vectors u, v to be u v. The distance version of the Triangle Inequality is as follows. If u, v, w are the vertices of a triangle in an inner product space V then u w u v + v w. It follows from the length version as u w = (u v) + (v w). If we take u, v, w to be vertices of a triangle in the Euclidean plane this gives the geometric version of the Triangle Inequality... Orthogonality It is not possible to define angles in a general inner product space, because inner products need not be real. But in any Euclidean space we can define these geometrical concepts even if the vectors have no obvious geometric significance.
4 Now we can use the Cauchy Schwarz inequality to define the angle between vectors. If u, v are nonzero vectors the angle between them is defined to be cos u v u. v. The Cauchy u v Schwarz inequality ensures that u. v lies between and. The angle between the vectors is / if and only if u v =. Example 9: Suppose we define the inner product between two continuous functions by u v / Solution: u u( x) v( x) dx. If u(x) = sin x and v(x) = x find the angle, between them in degrees. u v / x ux / / xsin xdx / = =. sin sin x x cos x (integrating by parts) xdx cos x = dx x / = sin x 4 = 4 which is approximately Hence u(x) is approximately.886. v x vx / x x = dx / = which is approximately Hence v(x)} is approximately.66. So if the angle (in degrees) between these two functions is then cos.886* Hence NOTE: Measuring the angle between two functions in degrees is rather useless and is done here only as a curiosity. By far the major application of angles in function spaces is to orthogonality. This is a concept that is meaningful for all inner product spaces, not just Euclidean ones. Two vectors in an inner product space are orthogonal if their inner product is zero. The same definition applies to Euclidean spaces, where angles are defined and there orthogonality means that either the angle between the vectors is / or one of the vectors is zero. So orthogonality is slightly more general than perpendicularity. A vector v in an inner product space is a unit vector if v =.
5 We define a set of vectors to be orthonormal if they are all unit vectors and each one is orthogonal to each of the others. An orthonormal basis is simply a basis that is orthonormal. Note that there is no such thing as an orthonormal vector. The property applies to a whole set of vectors, not to an individual vector. Theorem 4: An orthonormal set of vectors {v,, v n } is linearly independent. Proof: Suppose v + + n v n =. Then v + + n v n v r = for each r. But v + + n v n v r = v v r + + n v n v r = r v r v r since v r is orthogonal to the other vectors in the set = r since v r v r = v r =. Hence each r =. Because of the above theorem, if we want to show that a set of vectors is an orthonormal basis we need only show that it is orthonormal and that it spans the space. Linear independences come free. Another important consequence of the above theorem is that it is very easy to find the coordinates of a vector relative to an orthonormal basis. Theorem 5: If,,..., n is an orthogonal basis for the inner product V, and v V, then x v x v i where xi. i x n Proof: Let v = x + x x n n. Then v i x j j i j = x j j since the j i are mutually orthogonal Example : Consider R as an inner product space with the usual inner product. Show that the set,,,,,,,, is an orthonormal basis for R. Solution: They are clearly mutually orthogonal and, since = they are all unit vectors. Hence they are linearly independent and so span a dimensional subspace of R. Clearly this must be the whole of R. Example : Find the coordinates of (, 4, 5) relative to the above orthonormal basis. Solution: (, 4, 5) (/, /, /) = 7 (, 4, 5) (/, /, /) = (, 4, 5) (/, /, /) =. Hence the coordinates are (7,, ). In other words, (, 4, 5) = 7,,,,. Example : In C as an inner product space with the inner product (x, y ) (x, y ) = x x + y y show that the vectors ( i, 4i) and ( 4i, /5 + /5 i) are orthogonal. Use them to find an orthonormal basis for C.
6 Solution: ( i)( + 4i) + ( 4i)(/5 /5 i) = + 5i 5i =. ( i, 4i) = = and ( 4i, /5 + /5 i) = =. Hence { ( i, 4i), ( 4i, /5 + /5 i)} is an orthonormal basis. Theorem 6 (GramSchmidt): Every finitedimensional inner product space V has an orthonormal basis. Proof: We prove this by induction on the dimension of V. If dim(v) = then the empty set is an orthonormal basis. Suppose that every vector space of dimension n has an orthonormal basis of V and suppose that V is a vector space of dimension n +. Let {u,, u n+ } be a basis for V and let U = v,, v n. By the induction hypothesis U has an orthonormal basis v,, v n. Define w = u n+ u n+ v v u n+ v n v n. Then for each i, w v i = u n+ v i u n+ v i v i v i, since v j v i = when i j = u n+ v i v v i since v i v i = =. Hence w is orthogonal to each of the v i. But it may not be a unit vector, but it is nonzero. So we may divide it by its length without affecting orthogonality. So we define v n+ = w w and so obtain an orthonormal basis {v,, v n, v n+ } for V. In practice it is inconvenient to normalise the vectors (divide by their length) as we go, because we will have to carry these lengths along into our subsequent calculations. It s much easier to produce an orthogonal basis and then to normalise at the end. GRAMSCHMIDT ALGORITHM {u, u,..., u n } is a given basis; {v, v,..., v n } is an orthogonal basis; {w, w,..., w n } is an orthonormal basis. () LET v = u. () FOR r = TO n, LET v r = u r u r v v v u r v v v... u r v r v r v r. Multiply by a convenient factor to remove fractions. (4) FOR r = TO n, LET w r = v r v r. Example : Find an orthonormal basis for V = (,,, ), (,,, 4), (,,, ). Solution: u = basis v = orthogonal basis v w = orthonormal basis (,,, ) (,,, ) (,,, ) (,,, 4) (,,, ) 5 (,,, ) 5 (,,, ) (, 6, 6, ) 6 (6,, 8, ) 4
7 WORKING: v = u u v v v = (,,, 4) 4 (,,, ) = (,,, 4) 5 (,,, ). Multiply by, so now v = (, 4, 6, 8) 5(,,, ) = (,,, ). v = u u v v v u v v v = (,,, ) (,,, ) (,,, ). Multiply by, so now v = (,,, ) (5, 5, 5, 5) + (,,, ) = (, 6, 6, ). Example 4: Let V be the function space, x, x made into a Euclidean space by defining u x vx uxvx dx. Find an orthonormal basis for V. Solution: u = basis v = orthogonal basis v w = orthonormal basis x x (x ) x 6x 6x (6x 6x + ) WORKING: u x v x x dx x. x dx x v v (x) = u (x) u (x) v (x) v (x) v (x) = x. Multiply by, so now v (x) = x. u x v x x x. 4 ux vx x x dx x x dx x x 4 vx x dx 4x 4x dx x x x. v (x) = u (x) u (x) v (x) v (x) v (x) u (x) v (x) v (x) v (x) = x /6 / (x ) = x (x ) Multiply by 6, so now v (x) = 6x (x ) = 6x 6x vx 6 x 6x dx 6x 7x 48x x dx x 8x 6x 6x x
8 .4. Fourier Series The most important applications of inner product spaces involves function spaces with the inner product defined by means of an integral. Fourier series are infinite series in an infinite dimensional function space. However it is not appropriate here to give more than a cursory overview because to discuss them properly requires not only a good knowledge of integration, but a deep understanding of the convergence of infinite series. For any positive integer n the functions, cos nx and sin nx are periodic, with period. [This does not mean that they do not have smaller period, but simply that for each of them f(x + ) = f(x) for all x.] Take the space T spanned by all of these functions. Define the inner product on T as So T =, cos x, cos x,..., sin x, sin x,... u ( x) v( x) u( x) v( x) dx. T is an infinite dimensional vector space. Clearly, for every function f(x) T, f(x + ) = f(x). If f(x) is a continuous function for which f(x + ) = f(x) we may ask whether f(x) T. The answer is usually no. Such an f(x) may not be a linear combination of, cos x, cos x,..., sin x, sin x,... But remember that a linear combination is a finite linear combination. It may well be that f(x) can be expressed as an infinite series involving these functions. That is we might have f(x) = a + a cos nx b sin nx n n n. Such a series is called a Fourier series, named after the French mathematician Joseph de Fourier [768 8]. Of course, for this to make sense we would need this series to converge, which why we need to know a lot about infinite series in order to study Fourier series. But suppose we limit the values of n. Let T N =, sin x, sin x,..., cos x, cos x,... We can show that these N + functions are linearly independent. orthogonal. So T N is a N + dimensional Euclidean space. For n >, cos nx cos nx dx = and sin x = sin In fact, they are mutually nx dx =. Clearly = dx =. (Remember that here is not the absolute value but rather the length of the function.) N Fx By theorem 5, if F(x) = a + an cos nx bn sin nx then a = = F ( x) dx and, if n >, a n = F x cos nx cos nx n = F ( x)cos nx dx and b n = F x sin nx sin nx = F ( x)sin nx dx. A function in T N must be continuous and have period. But by no means does every such function belong to T N. However if F(x) is continuous and has period then it can be approximated by a function in T N, with the approximation getting better as N becomes larger. Even functions with period having some discontinuities can be so approximated. (We won t go into details here as to the precise conditions, or how close the approximation will be.) where a = If F (N) (x) is the approximation to F(x) in T N then F (N) (x) = a + a cos nx b sin nx F N ( x) dx and, if n >, a n = N n F N ( x)cos nx dx and b n = n F N ( x)sin nx dx. Now it can be shown that generally these integrals can be approximated by the corresponding integrals with F (N) replaced by F(x). Letting N it can be shown that if F[x] can be expressed as n a n n then a = a Fourier series a + cos nx b sin nx F ( x) dx and, if n >, n 6
9 a n = F ( x)cos nx dx and b n = F ( x)sin nx dx. Example 5: Find the Fourier series for the function F(x) = x if x / F(x) = x if / x / F(x) = x F(x + ) = F(x) for all x Answer: The solution involves a fair bit of integration by parts and, since this is not a calculus course, we omit the details and simply give the answer. 4 sin x sin x sin 5x F(x) =. 5 / / / /.5. Orthogonal Complements In R the normal to a plane through the origin is a plane through the origin. Every vector in the plane is orthogonal (i.e. perpendicular if they are nonzero) to every vector along the line. The line and the plane are said to be orthogonal complements of one another. The orthogonal complement of a subspace U, of V is defined to be: U = {v V u v = for all u U}. Intuitively it would seem that U should be U, but there are examples where this is not so. However for finitedimensional subspaces it is true. This follows from the following important theorem. Theorem 7: If U is a finitedimensional subspace of the vector space then U is also subspace of V and V = U U. Proof: () U is a subspace of V. Let v, w U and let u U. Then u v + w = u v + u w = + =. Hence v + w U T. Let v U and let be a scalar. Then u v =u v =. =. Hence v U and so we have shown that U is a subspace. () U U =. 7
10 Suppose v U U. Then v is orthogonal to itself, and so v v =. By the axioms of an inner product space this implies that v =. () V = U + U. Let u, u n be an orthonormal basis for U. Let v V, u = v u u v u n u n and let w = v u. Then for each i, w u i = v u i v u i u i u i since u j u i = if i j = v u i v u i since u i u i = =. Hence w U. Clearly u U. So v = u + w U + U. Theorem 8: If U is a subspace of a finitedimensional vector space then U = U. Proof: Suppose u U and let v U. Then u v =. Hence v u =. Since this holds for all v U, and so u U. So it follows that U U. Now V = U U = U U so dim U = dim U. Hence U = U. EXERCISES FOR CHAPTER Exercise : If v = (x, y ) and v = (x, y ) define u v = x x x y x y + 5y y and [u v] = x x + x y + x y + y y. Show that under one of these products R becomes a Euclidean space and under the other it is not a Euclidean space. Exercise : Find an orthonormal basis for (,, ), (,, 5). Exercise : Find an orthonormal basis for (,,,, ), (,,,, ), (,,,, ), (,,,, ). Exercise 4: Find an orthonormal basis for the function space, x, x where u x vx uxvx dx. Exercise 5: Find an orthonormal basis for the function space, x, sin x where / x vx uxvx u dx. Exercise 6: Find the orthogonal complement for (,, 6), (,, ) in R. Exercise 7: Find the orthogonal complement of (,,, ), (,,, ) in R 4. Exercise 8: Find the orthogonal complement of, x in the vector space, x, x, where u ( x) vx is defined to be u xvx dx. 8
11 SOLUTIONS FOR CHAPTER Exercise : Axioms (), (), () are easily checked for both products. The simplest way to check them is to let A = and B =. Then u v = uav T and [u v] = ubv T. It is now very simple to check these first three axioms. (4) If v = (x, y) then v v = x 4xy + 5y = (x y) + y for all x, y. If v = (x, y) then [v v] = x + 4xy + y = (x + y) y. When x = and y = this is negative. Hence under the product [u v] R is not a Euclidean space. (5) If v v = then x = y and y = so v =. Hence under the product v v, R is a Euclidean space. Exercise : u = basis v = orthogonal basis v w = orthonormal basis (,, ) (,, ) (,, ) (,, 5) (,, 6) 8 (,, 6) 8 WORKING: v = u u v v v = (,, 5) (,, ) = (,, 5) (,, ) = (,, 6). Exercise : u = basis v = orthogonal basis v w = orthonormal basis (,,,, ) (,,,, ) (,,,, ) (,,,, ) (,, 4,, ) 8 (,, 4,, ) 8 (,,,, ) (, 4, 4, 6, ) 7 (, 4, 4, 6, ) 7 4 (,,,, ) (9, 49, 56, 49, ) 6958 (9, 49, 56, 49, ) 6958 WORKING: v = u u v v v = (,,,, ) 4 (,,,, ). Multiply by 4. Now v = (4,, 4, 4, 4) (,,,, ) = (,, 4,, ). v = u u v v v u v v v = (,,,, ) 4 (,,,, ) 8 (,, 4,, ). Multiply by 8. Now v = (8, 8, 8,, 8) (,,,, ) (, 9,,, ) = (4, 6, 6, 4, 4). Divide by 4. Now v = (, 4, 4, 6, ). v 4 = u 4 u 4 v v v u 4 v v v u 4 v v v = (,,,, ) (,,,, ) 8 (,, 4,, ) 7 (, 4, 4, 6, ). Multiply by 4. Now v 4 = (4, 4, 4, 4, ) (,,,, ) (5, 45, 6, 5, 5) (6, 4, 4, 6, 6) = (9, 49, 56, 49, ). 9
12 Exercise 4: u = basis v = orthogonal basis v w = orthonormal basis x x (x ) x x x + (x x + ) WORKING: / u x v x x dx = x =. v (x) = u (x) u (x) v (x) v (x) v (x) = x /. = x. Multiply by, so now v (x) = x. v x x u dx = x x 4 9 dx 9 / = x 8x 9 = 8 4 =. x v x x dx = x =. 4x x v x x x dx u / = x dx x dx = x 5 = 6 5 = 5. 5 / x v (x) = u (x) u (x) v (x) v (x) v (x) u (x) v (x) v (x) v (x) = x / /5. / (x ) 4
13 = x 5 (x ) = x 6 5 x = x 6 5 x +. Multiplying by we now take v (x) = x x +. v (x) = x x dx = x 44x 9 4x / 6x 7 x dx = x 4x 9 4x / 7 x dx = 5/ / x x x x x = = =. Exercise 5: u = basis v = orthogonal basis v w = orthonormal basis WORKING: x x sin x sin x / / v x dx = / x v x x = / u xdx = x = v (x) = u (x) u (x) v (x) v (x) v (x) = x /4 /. = x. Multiply by, so now v (x) = x. v / x x dx / = 4x 4x dx x 6 8 sin x 8
14 4 = x x x = 6 = 6. / x v x x dx u sin / = =. cos x / / x v x x x dx u sin / = xsin x dx / sin = cos x x dx / / / x cos x dx x dx sin / / = + sin x + cos x = =. v (x) = u (x) u (x) v (x) v (x) v (x) u (x) v (x) v (x) v (x) = sin x..(cos x sin x) = sin x. Multiplying by we now take v (x) = sin x. / v (x) = sin x dx / = sin x 4 sin x 4 / dx = sin dx 4 sin x dx + 4 dx / = cos / / = sin x = 4 + / / x dx 4 sin x dx + 4 dx / x + / 4 cos x + 4
15 = 4 4. = 8 Exercise 6: First Solution: Let u = (,, ) and u = (,, 6). Suppose (x, y, z) is orthogonal to both u and u. Then x + y + z = and x + y + 6z =. so x=, z = k, y = k. 6 4 Hence the orthogonal complement is (,, ). Second Solution: We could take, as a third vector (,, ), being outside of the space spanned by a and b, and use the Gram Schmidt process. However we are content with an orthogonal basis. u = basis v = orthogonal basis v (,, ) (,, ) 9 (,, 6) (5,, 4) 45 (,, ) (,, ) WORKING: v = u u v v v = (,, 6) 9 9 (,, ). Multiply by 9 to get the new v to be v = 9(,, 6) 9(,, ) = (8, 7, 54) (8, 9, 8) = (, 8, 6). Perhaps it would now be a good idea to divide by 4 to get a new v as v = (5,, 4). u v = 5 and u v =. v = u u v v v u v v v = (,, ) 5 9 (,, ) 45 (5,, 4) Multiply by 45 to get a new v as v = (45, 45, 45) (5, 5, 5) (5,, 4) = (, 8, 9). Divide by 9 to get a new v as v = (,, ). Hence the orthogonal complement is (,, ). Third Solution: A third method, that only works for R, is to simply find u u. i j k u u = = (6 6)i ( 4)j + (6 )k = (, 8, 4). So the orthogonal complement is 6 (, 8, 4) = (,, ). You can make up your own mind as to which is the easiest method! Exercise 7: Here we cannot use the vector product. Suppose (x, y, z, w) is orthogonal to both vectors. Then we have a system of two homogeneous linear equations that is represented by 4
16 44. So w = h, z = k, y = h, x = k giving the vector (k, h, k, h). Taking h =, k = and h =, k =, we get a basis for the orthogonal complement, which is (,,, ), (,,, ). Exercise 8: dx cx bx a. = x c x b ax = a + b + c and dx x cx bx a. = 4 4 x c x b x a = a + b + c 4. Hence w(x) = a + bx + cx is orthogonal to both and x if a + b + c = and a + b + c 4 =. We solve the homogeneous system This gives c = k, b = k, 6a = 5k. Take k = 6. Then a = 5, b = 6, c = 6 and hence w(x) = 5 + 6x + 6x. Hence the orthogonal complement is 6x + 6x 5.
Inner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationSection 4.4 Inner Product Spaces
Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More information17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function
17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 6 Woman teaching geometry, from a fourteenthcentury edition of Euclid s geometry book. Inner Product Spaces In making the definition of a vector space, we generalized the linear structure (addition
More informationOrthogonal Projections and Orthonormal Bases
CS 3, HANDOUT A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationMath 4310 Handout  Quotient Vector Spaces
Math 4310 Handout  Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationDeterminants, Areas and Volumes
Determinants, Areas and Volumes Theodore Voronov Part 2 Areas and Volumes The area of a twodimensional object such as a region of the plane and the volume of a threedimensional object such as a solid
More informationMATH1231 Algebra, 2015 Chapter 7: Linear maps
MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. FiniteDimensional
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
More informationMAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationVectors, Gradient, Divergence and Curl.
Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use
More informationVector Math Computer Graphics Scott D. Anderson
Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
More informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More informationLinear Algebra: Vectors
A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector
More informationNOV  30211/II. 1. Let f(z) = sin z, z C. Then f(z) : 3. Let the sequence {a n } be given. (A) is bounded in the complex plane
Mathematical Sciences Paper II Time Allowed : 75 Minutes] [Maximum Marks : 100 Note : This Paper contains Fifty (50) multiple choice questions. Each question carries Two () marks. Attempt All questions.
More informationCS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationChapter 20. Vector Spaces and Bases
Chapter 20. Vector Spaces and Bases In this course, we have proceeded stepbystep through lowdimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 51 Orthonormal
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More informationBX in ( u, v) basis in two ways. On the one hand, AN = u+
1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
More informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More informationMAT 242 Test 3 SOLUTIONS, FORM A
MAT Test SOLUTIONS, FORM A. Let v =, v =, and v =. Note that B = { v, v, v } is an orthogonal set. Also, let W be the subspace spanned by { v, v, v }. A = 8 a. [5 points] Find the orthogonal projection
More information2. THE xy PLANE 7 C7
2. THE xy PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
More informationChapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors
Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationRESULTANT AND DISCRIMINANT OF POLYNOMIALS
RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8
Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More information1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0
Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are
More informationHøgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver
Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin email: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider twopoint
More information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More information1 3 4 = 8i + 20j 13k. x + w. y + w
) Find the point of intersection of the lines x = t +, y = 3t + 4, z = 4t + 5, and x = 6s + 3, y = 5s +, z = 4s + 9, and then find the plane containing these two lines. Solution. Solve the system of equations
More information18.06 Problem Set 4 Solution Due Wednesday, 11 March 2009 at 4 pm in 2106. Total: 175 points.
806 Problem Set 4 Solution Due Wednesday, March 2009 at 4 pm in 206 Total: 75 points Problem : A is an m n matrix of rank r Suppose there are righthandsides b for which A x = b has no solution (a) What
More informationElementary Linear Algebra
Elementary Linear Algebra Kuttler January, Saylor URL: http://wwwsaylororg/courses/ma/ Saylor URL: http://wwwsaylororg/courses/ma/ Contents Some Prerequisite Topics Sets And Set Notation Functions Graphs
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More information28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z
28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition
More informationCHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.
More informationReview Jeopardy. Blue vs. Orange. Review Jeopardy
Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 03 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?
More informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More information