The Dot Product of Two Vectors in n

Transcription

1 These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (rd edition) These notes are intended primarily for in-class presentation and should not be regarded as a substitute for thoroughly reading the textbook itself and working through the exercises therein Suppose that The Dot Product of Two Vectors in n u u u 2 and v v v 2 are two vectors in 2 We define the dot product (also called the inner product or the scalar product) of u and v to be u v u T v Thus u v u T v u u 2 v v 2 u v u 2 v 2 Note that u v is actually a matrix However we will adopt the convention that u v is a scalar (equal to the single entry in this matrix) For example if u and v 7 then u v 7 2 Clearly this definition can be extended to n for any positive integer n: If u v u u 2 and v v 2 u n v n are any two vectors in n then we define u v u v u 2 v 2 u n v n When convenient we will simply write u v u T v (even though technically u v is a scalar and u T v is a matrix)

2 Example Let u 6 8 v and w Compute u v v u u v u v u v u u u v w and u v u w 2

3 Basic Properties of the Dot Product If u and v are any two vectors in n then u v v u 2 If u and v are any two vectors in n and c is any scalar then cu v u cv cu v If u v and w are any three vectors in n then u v w u v u w If u is any vector in n then u u Also u u if and only if u n

4 Length and Distance For any vector u n we define the length (also called the norm or the magnitude) ofu to be u u u Thus if u u u 2 then u n u u u u 2 u 2 2 u n 2 Example Find the length of the vector u

5 If u and v are any two vectors in n then we define the distance between u and v to be u v Example Find the distance between the vectors u 8 and v 9

6 Basic Properties of Length and Distance If u is any vector in n then u Also u if and only if u n 2 If u is any vector in n and c is any scalar then cu c u If u and v are any two vectors in n then u v v u If u and v are any two vectors in n then u v uv (This is called the Cauchy Schwarz Inequality) If u and v are any two vectors in n then u v u v (This is called the Triangle Inequality) 6

7 A unit vector is a vector of length Ifv is any vector in n with v n then it is easy to find a unit vector that points in the same direction as v: In fact the vector u v v has length and is a positive scalar multiple of v Example Find a unit vector that points in the same direction as the vector u 6 8 7

8 Orthogonality of Vectors Two vectors u and v in n are said to be orthogonal (or perpendicular) to each other if u v This definition of orthogonality is inspired by what we can visualize in 2 or : Two non zero vectors u and v in 2 (or ) are perpendicular to each other if and only if u v u v and this is true if and only if u v Example Show that the vectors u 9 and v are orthogonal to each other and show that the vectors 2 u and v 6 are not orthogonal to each other 8

9 Example Describe the set of all vectors in 2 that are orthogonal to the vector 9 u 9

10 Example Describe the set of all vectors in that are orthogonal to the vector u 7

11 Example Let W Span Note that W is a two dimensional subspace of (a plane in that passes through the origin in ) 6 Describe the set of all vectors in that are orthogonal to every vector in W

12 Definition If W is a subspace of n we define the orthogonal complement of W denoted by W to be the set of all vectors in n that are orthogonal to all of the vectors in W Theorem Suppose that W is a subspace of n Then: W is a subspace of n 2 W W n dimw dimw n and if B W is a basis for W and B W is a basis for W then B W B W a basis for n Remark If W and U are any two subspaces of n such that W U n and such that the union of a basis for W and a basis for U is a basis for n then we say that n is the direct sum of W and U and we write n W U Thus the above theorem tells us that if W is any subspace of n then n W W We remark that but that n W U U W U W n W U Example If W Span 9 then W is a subspace of 2 In fact W is a line passing through the origin in 2 The subspace W consists of all vectors x x x 2 2 such that 9x x 2 or to write this in a different way 9 x x 2 Since 9 ~ 9 2

13 we see that x x x 2 t 9 t t 9 t 9 Therefore W Span 9 Note that W W Span Example If W Span 7 then W is a subspace of In fact W is a line passing through the origin in The subspace W consists of all vectors x x x 2 2 x

14 such that x x 2 7x or to write this in a different way 7 x x 2 x Since 7 ~ 7 we see that x x x 2 t 7 s t t s 7 t s 7 x s Therefore 7 W Span Note that 7 W W Span 7 Example If W Span 6 then W is a subspace of In fact W is a plane passing through the origin in The subspace W consists of all vectors x x x 2 x 2 such that

15 x x 2 6x x x or to write this in a different way 6 x x 2 x Since 6 ~ 2 we see that x t x x 2 x 2 t t t 2 t 2 Therefore W Span 2 Note that W W Span 2 6 Theorem If A is any m n matrix then rowa nula and n rowa nula Also cola nula T and m cola nula T Proof Suppose that r rowa and n nula Then r A T x for some vector x m and An m Now note that r n r T n A T x T n x T An x T An x T m This shows that every vector in rowa is orthogonal to every vector in nula and hence that rowa nula Now suppose that x rowa Then r x for all vectors r rowa Thus

16 r r x Ax r 2 x r 2 x m r m r m x which shows that x nula and hence that rowa nula We conclude that rowa nula and hence that n rowa nula To verify the second assertion of the theorem note that cola rowa T nula T Example In order to illustrate the above theorem suppose that A is the matrix A We will show that rowa nula and that cola nula T First note that A 7 9 ~ ~ which shows that rowa Span We also see that nula consists of all vectors x such that x t x x 2 x t t t t 2 Therefore nula Span Now observe that rowa nula 2 6

17 Next note that 2 2 A T 6 7 ~ 9 9 which shows that cola rowa T Span 2 and nula T Span We observe that cola nula T 2 As a double check observe that the row reduction that was performed earlier: A 7 9 ~ shows that 2 6 cola Span 7 from which it can still be observed that cola nula T 7

18 Extending the Idea of Orthogonality to Vector Spaces Other Than n Definition Let V be a vector space An inner product for V is a binary function : V such that If u and v are any two vectors in V then u vv u 2 If u and v are any two vectors in V and c is any scalar then cu vu cvcu v If u v and w are any three vectors in V then u v w u v u w If u is any vector in V then u u and u u if and only if u V Example The dot product is an inner product for n Example For the vector space C (consisting of all continuous functions f : ) we can define an inner product as follows: fg fxgxdx To verify that this is indeed an inner product for C note that: If f and g are any two functions in C then fg fxgxdx gxfxdx gf 2 If f and g are any two functions in C and c is any scalar then cfg cfxgxdx c fxgxdx fcg fxcgxdx c fxgxdx cfg c fxgxdx Thus cfg fcg cfg If f g and h are any three functions in C then fg h fxgx hxdx fxgx fxhxdx fxgxdx fxhxdx fg fh If f is any function in C then fx 2 dx ff fxfxdx 8

19 (because fx 2 for all x Also fx 2 dx if and only if fx for all x which means that ff if and only if f is the zero vector (that is the zero function) in C Definition If V is a vector space with an inner product then we define the norm of any vector v Vtobe v vv If u then we say that u is a unit vector As before if we are given a vector v V then a unit vector in the same direction as v is u v v Example Let V be the function space C with inner product as defined above Consider the following functions which are all members of C : f t f t cost g t sint f 2t cos2t g 2t sin2t f t cost Find the norms of these functions and find a unit function corresponding to each of them (In other words for each of the above functions f i find a function i such that i and i is a scalar multiple of f i and for each of the functions g i find a function i such that i and i is a scalar multiple of g i ) Solution f 2 f f fx 2 dx dx 2 9

20 shows that f 2 Therefore a unit function in the same direction as f is 2 f This is the constant function defined by t 2 for all t Next we find : f 2 f f fx 2 dx cos 2 xdx shows that f Therefore a unit function in the same direction as f is f This is the function defined by t cost for all t Next we find : g 2 g g gx 2 dx sin 2 xdx shows that g Therefore a unit function in the same direction as g is g This is the function defined by t sint for all t Next we find 2 : 2

21 f 2 2 f 2 f 2 f2x 2 dx cos 2 2xdx shows that f 2 Therefore a unit function in the same direction as f 2 is 2 f 2 This is the function defined by 2t cos2t for all t Does there seem to be a pattern here? There is In fact it can be shown that t 2 nt cosnt n nt sinnt n Definition If V is a vector space with an inner product and B is a basis for V then B is called an orthonormal basis if every vector in B is a unit vector and any two vectors in B are orthogonal to each other Example The standard basis E e e 2 e n is an orthonormal basis for n Example The basis B is an orthonormal basis for 2 2

22 Theorem If V is a vector space with an inner product and with orthonormal basis B b b 2 b n and if v is any vector in V then vb v B vb 2 vb n In other words v vb b vb 2b 2 vb nb n Proof Since B is a basis for V we know that v can be written as v c b c 2 b 2 c n b n Note that vb b v b c b c 2 b 2 c n b n b c b b c 2 b 2 b c n b n c b b c 2b b 2 c nb b n c b 2 c Likewise it can be shown that c 2 vb 2c n vb n Example The basis B b b 2 is an orthonormal basis for 2 Find the coordinates of the vector v relative to this basis Solution and Thus vb vb 2 2 v 2b b 2 22

23 The most interesting and powerful applications of these ideas come into play when we are dealing with function spaces which are infinite dimensional vector spaces We cannot study this in great detail in this course However we can get a feel for the basic ideas Here is our main example: As we saw in an earlier example every one of the functions t 2 nt cosnt n nt sinnt n has norm It is also true that any two functions in this set are orthogonal to each other For example t tdt and cost sint dt (because the integrand is an odd function) t tdt cost cost costcostdt dt (is seen by using integration by parts) Thus the set of functions 2 2 is an orthonormal set of functions in the function space C Motivated by the previous theorem (which pertains to finite dimensional vector spaces) we are led to ask the following question: Given any function f C is it true that f f f f f 2 2 f 2 2? In other words is it true that ft f 2 f cost f sint f 2 cos2t f 2 sin2t for all t? Under certain (rather small) restrictions on the function f it can be proved that the answer to this question is Yes We cannot delve into all of the details which would require a course in advanced calculus at the minimum Let us just point out that the 2

24 infinite trigonometric series on the right is called the Fourier series of the function f and that it is often true that f is in fact equal to its Fourier series The coefficients f 2 f f f 2 f 2 that appear in the Fourier Series are called the Fourier coefficients of f Example The function fx x 2 is a function in C Find the first ten Fourier coefficients of f and compare the graphs of f with the graphs of the truncated Fourier series of f consisting of the first ten terms of the series Solution f 2 x f f f 2 f 2 f f f f f x 2 cosx x 2 x 2 dx 2 dx sinx dx cos2x dx x 2 cosx dx 9 x 2 cosx dx x 2 cosx dx 2 A truncated Fourier series for fx x 2 is thus 2 cosx cos2x 9 cosx cosx 2 cosx The graph of this function and the graph of fx x 2 are shown below 2

25 x 2