Algebra s Tarleton State University January 6, 2012
Spanning a
Common Notation Definition N = {1, 2, 3,...} is the set of natural numbers. W = {0, 1, 2, 3,...} is the set of whole numbers. Z = { 3, 2, 1, 0, 1, 2, 3,...} is the set of integers. { a } Q = b a, b Z, b 0 is the set of rational numbers. R is the set of real numbers. Let i = 1. Then C = {a + bi a, b R} is the set of complex numbers. The set of polynomials whose degree is less than or equal to n is P n = {a 0 + a 1 t + a 2 t 2 + + a n t n a 0, a 1,..., a n R}. M m n is the set of all (real valued) matrices of size m n (m rows and n columns).
Defined Definition Let set F be a set of elements, scalars, with operations, (addition & multiplication). A vector space is a set V of elements called vectors, with vector addition,, and scalar multiplication,, satisfying the ten axioms: Additive Properties 1. Closure for Addition If u, v V, then u v V. 2. Commutativity of Addition If u, v V, then u v = v u. 3. Associativity of Addition If u, v, w V, then (u v) w = u (v w). 4. There exists 0 V such that for all u V, u 0 = u. 5. For all u V, there exists at least one element ũ V such that ũ u = 0. Fact: ũ is unique! Let ũ = u.
Definition Continued Definition (Continuation of Definition) Product Properties 1. Closure for Scalar Vector Product If α F and u V, then α u V. 2. Distributive Law: Scalar Times Vectors For any α F and u, v V, we have α (u v) = (α u) (α v). 3. Distributive Law: Scalar Sum Times Vector For any α, β F and u V, we have (α β) u = (α u) (β u). 4. Associativity of Scalar Vector Product If α, β F and u V, then α (β u) = (α β) u. 5. Multiplicative Identity: Unit Scalar Times Vector There exists 1 F such that for each u V, 1 u = u.
Important Remarks! Remark Before you talk about a vector space, you must define the set F, the operations on F :,,, and the set V, and the operations on V :,. To make our life easier, the set of scalars, F, will usually be R, the real numbers, the usual addition of real numbers, and the usual multiplication of real numbers. For those of you who don t have bold built in to their handwriting, then you might want to use an arrow above vectors so that we can distinguish them from scalars!
R n Definition An n-tuple of real numbers is a list of n real numbers, x 1, x 2,..., x n in an array (x 1, x 2,..., x n ). An n-tuple is often called a point or a vector. We define R n to be the set of all n-tuples of real numbers, that is R n = {(x 1, x 2,..., x n ) x 1, x 2,..., x n R}. Example R 2 = {(x 1, x 2 ) x 1, x 2 R} is the two dimensional plane.
Operations on R n Definition Let x, y R n and α R. We assume that x = (x 1, x 2,..., x n ) and y = (y 1, y 2,..., y n ). Vector Addition We define vector addition by x + y = (x 1 + y 1, x 2 + y 2,..., x n + y n ). Scalar Multiplication We define scalar multiplication by αx = (αx 1, αx 2,..., αx n ). Zero Vector The zero vector is 0 = (0, 0,..., 0). Unit Scalar The unit scalar is 1 R.
R n is a The set R n with the definitions for vector addition as, scalar multiplication for, zero vector, unit scalar as well as the usual addition and multiplication on F = R define a vector space. Proof: Additive Properties 1. Closure for Addition If u, v V, then u v V. 2. Commutativity of Addition If u, v V, then u v = v u. 3. Associativity of Addition If u, v, w V, then (u v) w = u (v w). 4. There exists 0 V such that for all u V, u 0 = u. 5. For all u V, there exists at least one element ũ V such that ũ u = 0. Fact: ũ is unique! Let ũ = u.
Proof Continued Product Properties 1. Closure for Scalar Vector Product If α F and u V, then α u V. 2. Distributive Law: Scalar Times Vectors For any α F and u, v V, we have α (u v) = (α u) (α v). 3. Distributive Law: Scalar Sum Times Vector For any α, β F and u V, we have (α β) u = (α u) (β u). 4. Associativity of Scalar Vector Product If α, β F and u V, then α (β u) = (α β) u. 5. Multiplicative Identity: Unit Scalar Times Vector There exists 1 F such that for each u V, 1 u = u.
Examples s Example In this example we assume that F = R with the usual addition or real numbers and multiplication or real numbers for and. The set V = M m n is a vector space with the usual matrix addition and scalar multiplication for and. The set V = P n is a vector space with the usual polynomial addition and scalar multiplication for and. (This is HWK) The set V = C (, ) of continuous, real valued functions (functions whose domain is R) is a vector space with the usual operations for function addition and scalar multiplication, that is for x R, f, g V, and α F, f g = (f + g)(x) and α f = αf (x).
Example Let s Do It! Let F = R, and V = R + = (0, ) the set of positive real numbers. Define u v = uv and α u = u α. Show that this is a vector space.
Example Let s Do It! Let F = R, and V = R. Define u v = u v and α u = αu. Show that this is not a vector space.
Example Example Let the set V = Q usual addition and scalar multiplication of numbers. If F = R, then V is not a vector space. If F = Q, then V is a vector space.
Important Properties for all s If V is a vector space with scalar set F, then (a) The vector 0 is unique. (This is HWK) (b) The additive inverse ũ is unique, so we name it u. (c) 0 u = 0 for all u V. (d) α 0 = 0 for all α F. (This is HWK) (e) If α u = 0, then either α = 0 or u = 0. (THis is HWK) (f) ( 1) u = u for all u V.
Homework Pages: 196-197 1, 2. 6, 7, 8, 16, 19, 21, 26 Use Mathematica for problem 26. Print out your code. Note: In Mathematica, a vector is written using curly braces, so the vector v = (1, 2, 3) is typed v = {1, 2, 3} Addition works as you would expect. So, if v, 2 are vectors of the same size, then addition is v + w Scalar multiplication is as you would expect, so to multiply vector v by 5 you can write 5 v
Definition Let V be a vector space with scalar set F and W a nonempty subset of V. If W is a vector space with respect to the operations in V and the same scalar set F, then W is a subspace of V. Example There are two trivial examples of subspaces of a vector space V. Namely, {0} (the zero subspace) and V!
Cool Theory Let V be a vector space with operations and and let W be a nonempty subset of V. The W is a subspace of V if and only if the following conditions hold: (a) If u, v W, then u v W. (b) If α F and u W, then α u W.
Example Let s Do It! Show that P 3 (set of all polynomials of degree 3) is a subspace of the set of all polynomials. Let W be the set of polynomials of degree 3. Show that this is not a subspace of the set of all polynomials.
I m Bored From now on, it I will normally use u + v rather than u v and αu rather than α u.
Combinations and Span Definition Suppose V is a vector space with scalars F. Let u 1, u 2,..., u n V and α 1, α 2,..., α n F. A linear combination of these vectors is n α i u i = α 1 u 1 + α 2 u 2 + + α n u n. i=1 Definition The span of a (nonempty) set of vectors, S, is the collection of all linear combinations of vectors in the given set and is denoted by Span(S).
Example Example 1 Consider the vector space R 3. Choose v = 2 and 3 0 w = 1. 1 Write the Span({v, w}) in set builder notation. What does this represent? (Cal III students should know this...)
Let V be a vector space with scalar set F and suppose S is a nonempty set of vectors of V. Then Span(S) is a subspace of V.
Example Definition Let e 1 = (1, 0,..., 0), e 2 = (0, 1, 0,..., 0),... e n = (0,..., 0, 1) R n. These vectors are called the standard unit vectors. Let s Do It! Suppose we consider R 2. Write the span of f = (1, 2)? What is the span of the standard unit vectors of R 2? (e 1 = (1, 0), e 2 = (0, 1))
Recall Some Matrix Theory Remark Suppose M = Show that Mv = x [ ] a b c M d e f 2 3. Let v = [ ] a + y d [ ] b + z e [ c f x y. z ]. How do you determine if there exists a vector v such that [ ] u Mv =? w Find the row reduced echelon form of the augmented matrix [ a b c u ] d e f w
Example Example Is the vector (1, 2, 3) in the span of the vectors (2, 0, 3), ( 4, 1, 2), (5, 4, 3)? What is the vector (1, 2, 3) in the span of the vectors (2, 0, 3), ( 4, 1, 2), (5, 4, 3)? Is the vector (1, 2, 3) in the span of the vectors (2, 0, 2), ( 4, 1, 2), (6, 4, 14)?
Kernel or Null of a Matrix Definition The kernel or null space or solution space of the matrix A is the set of all vectors x such that Let s Do It! Ax = 0. Let A M m n. Show that the null space of A is a subspace of R n.
All About LInes in R 2 Example Let v = (1, 2). Graph v, 2v, 2v. For all α R, what is αv = Span(v)? Let u = (3, 2). What is u + αv for α R? What is {t(u v) t R}? What about αu + (1 α)v for α R?
Tiny Definition Definition Let S V be a subset of a vector space V. If w V, then w + S = {w + y y S}. Remark Note that this is yet another abuse of the notation +. (In C++ terms, the operator + is overloaded.) The expression w + S is a vector plus a set while w + y is the sum of two vectors...very different uses of +!
Geometric Interpretation of Lines in R n Multiplication of a vector by all possible scalar generates a line passing through 0. That is, L 1 = {αu α R} = Span(u) is a line passing through 0 for any fixed u R n. If u, v R n are two vectors and v u, then a line through the points u and v is given by L 2 = {tu + (1 t)v t R} = {v + t(u v) t R} = v + {t(u v) t R}. Remark Notice that the line L 2 is really a translation by vector v of the line {t(u v) t R} which passes through the origin.
Parametric Representation of Lines Remark Recall that points on the line L 2 are given by x = v + t(u v) where t R and u v. Let w = u v 0. Then points on the line L 2 are given by x = u + tw where t R. Definition The parametric representation of a line through the points u and w 0 is given by x = x(t) = u + tw, where t R.
Parametric Representation of Lines Definition The parametric representation of a line through the points u 1 w 1 u = u 2 and w = w 2 0 is given by u 3 w 3 x = x(t) = u + tw where t R. It can also be written as x(t) = u 1 + tw 1 y(t) = u 2 + tw 2 z(t) = u 3 + tw 3.
Example Example Let u = (1, 2, 3) and v = (1, 0, 0). Write an equation for the line through these points. Graph the line. Does the point (0, 3, 2) lie on the line?
Tiny Theory Let L 1 = {u + tv t R} and L 2 = {w + tz t R} be two lines in R n. Then these lines are the same if and only if u w and v are multiples of z. That is, u w = βz and v = αz for suitable scalars α and β. Corollary Let L 1 = {u + tv t R} and L 2 = {w + tz t R} be two lines in R n. These two lines are parallel and distinct if there exists α such that v = αz and u w βz for all β. These two lines intersect if u w Span({z, v}).
Example Example Do the lines (1, 2, 3) + t(2, 2, 0) and t(1, 1, 0) intersect? Are they parallel and distinct?
Homework Pages: 196-197 23, 24 Pages: 205-209 2, 3, 5, 7, 13, 14, 15, 17, 19, 23, 24, 31, 33, 36
Recall: Combinations and Span Definition Suppose V is a vector space with scalars F. Let u 1, u 2,..., u n V and α 1, α 2,..., α n F. A linear combination of these vectors is n α i u i = α 1 u 1 + α 2 u 2 + + α n u n. i=1 Definition The span of a (nonempty) set of vectors, S, is the collection of all linear combinations of vectors in the given set and is denoted by Span(S).
Spanning Sets Definition Let S be a nonempty subset of a vector space V. If V = Span(S) then we say that S is a spanning set of V. Example [ ] [ ] 1 0 Consider R 2. Note that e 1 =, e 0 2 = R 1 2. Show that S = {e 1, e 2 } is a spanning set of R 2.
Example Let s Do It! Find a simple [ spanning ] set for M 2 3. Hint, consider the a b c matrix A = and write is as a linear d e f combination of six matrices.
Example Let s Do It! Find a simple spanning set for the solution space to the homogeneous system [ ] 1 1 1 u = 0. 1 2 0
Nice Any set of fewer than n vectors will not span R n
Homework Pages: 205-209 25, 27 Pages: 215-216 1, 2, 3, 4, 13, 17 In Mathematica there are two useful commands you should know: RowReduce[M], which performs Gaussian elimination to find the row-reduced form of true matrix M (NOT row-reduced echelon form!) Null[M] which finds the kernel or null space or solution space of a matrix M.
for Finite Sets Definition Let S = {x 1, x 2,..., x n } be a set of vectors from a vector space. S is linearly independent if then c 1 x 1 + c 2 x 2 + + c n x n = 0 c 1 = c 2 = = c n = 0. Otherwise, S is linearly dependent and there is a non-trivial solution to the equation c 1 x 1 + c 2 x 2 + + c n x n = 0.
Tiny Remark Remark If the set S is linearly independent, then none of the vectors can be expressed as a linear combination of the others. If the set S is linearly dependent, this really means that at least one of the vectors can be expressed as a linear combination of the others. That is, when c 1 x 1 + c 2 x 2 + + c n x n = 0 there is some c i 0 (so we can talk about c 1 i ) and hence x i = c 1 i j i c j x j.
Example Example Test the following vectors for linear independence. x 1 = (1, 0, 0), x 2 = (0, 1, 0), and x 3 = (0, 0, 1). x 1 = (1, 2, 3), x 2 = ( 2, 3, 0), and x 3 = (0, 7, 6).
Example Let s Do It! Test the following vectors for linear independence. x 1 = t 2 1, x 2 = t 2 + t, and x 3 = 2t + 2.
is Way Cool! Each vector in the span of a linearly independent set (in a vector space) has a unique representation as a linear combination of elements of that set.
A Tiny Recall from Matrix Algebra These properties of an m n matrix A are equivalent: 1. The rank (# of nonzero rows in reduced row echelon form) of A is less than n. 2. The reduced row echelon form of A has fewer than n nonzero rows. 3. The matrix A has fewer than n pivot positions. 4. At least one column in A has no pivot position. 5. There is at least one free variable in the system of equations Ax = 0. 6. Ax = 0 has at least one nontrivial solution.
A Tiny Recall from Matrix Algebra Corollary A homogeneous system of linear equations in which there are more variables than equations must have some nontrivial solutions.
Nice Any set of more than n vectors in R n is linearly dependent.
Vectors in a and Dependence Let {u 1,..., u m } be a linearly dependent set of at least two vectors in V. Then some vector in that list is a linear combination of preceding vectors in that list. If a vector space is spanned by some set of n vectors, then every set of more than n vectors in that space must be linearly dependent.
Making Bigger Independent Sets Let S be a linearly independent set in a vector space V. If x V and x Span(S), then S {x} is linearly independent.
in R n - The Quick Way to Test for Let S = {v 1, v 2,..., v n } be a set of n vectors in R n. Let A = [v 1 v 2... v n ]. Then S is linearly independent if and only if det(a) 0.
and Dependence of Subsets and Supersets Suppose V is a vector space. Let S 1 S 2 V. Then (a) If S 1 is in early dependent, so is S 2. (b) If S 2 is independent, so is S 1.
General Definition for Infinite Sets Definition Let S be a set of vectors from a vector space. S is linearly independent if combination implies every finite linear c 1 x 1 + c 2 x 2 + + c k x k = 0 c 1 = c 2 = = c k = 0. Otherwise, S is linearly dependent and there is a non-trivial solution to at least one equation c 1 x 1 + c 2 x 2 + + c k x k = 0.
Example of an Infinite Independent Set Example Let p j (t) = t j for j = 0, 1, 2, 3,.... The set {p j j = 0, 1, 2,...} is independent. Clearly, this set spans P, the set of all polynomials using coefficients from R. Also, P is a vector space using the usual polynomial addition and polynomial scalar multiplication.
Homework Pages: 215-216 15, 16 Pages: 226-228 1, 2, 7, 11, 13, 17, 18, 24, 27, 29 In Mathematica, the command Flatten[Transpose[M]] will return a m n matrix as a mn vector where we form a vector by using the columns of the matrix. If you use the command Flatten[M], then it will form a vector by using the rows of the matrix. Try it on a matrix so that you understand how it works.
Bases Definition Let V be a vector space. A linearly independent set that spans V is called a basis for V. Example In R n, then standard unit vectors e 1,..., e n for a basis.
Example Let s Do It! Do the columns of A form a basis for R 3? What about the columns of B? 1 2 3 A = 2 3 1 3 4 5 1 2 3 B = 2 3 1 3 4 4
Example Example Find a basis for the vector space spanned by the four polynomials p 1 (t) = t 2 + 4, p 2 (t) = t 1, p 3 (t) = t 3 + t 2, and p 4 (t) = t 2 + 2t + 2.
Matrix Theory The columns of any n n invertible matrix is a basis for R n.
Very Important Theory If a basis is given for a vector space, then each vector in the space has a unique expression as a linear combination of elements in that basis.
If a vector space has a finite basis, then all of its bases have the same number of elements. Definition A vector space is finite dimensional if it has a finite basis. Otherwise, it is infinite dimensional. Let V be a finite dimensional vector space with any basis B. Then the dimension of V is the number of elements in the basis, that is Dim(V ) = B.
Example Example A basis for R n is B = {e 1,..., e n }. Thus Dim(R n ) = n. A basis for M 2,2 is C = {[ ] 1 0, 0 0 Thus, Dim(M 2,2 ) = 4. [ ] 0 1, 0 0 [ ] 0 0, 1 0 [ ]} 0 0. 0 1
More Cool Theory The span of a set is not affected by removing from the set one element that is a linear combination of other elements of that set. If a set of n vectors spans an n-dimensional vector space, then the set is a basis for that vector space. In an n-dimensional vector space, every linearly independent set of n vectors is a basis. Proof: (Pick one independence or spanning)
Miscellaneous Cool Theory If a set of n vectors spans a vector space, then the dimension of that space is at most n. If a set of n vectors is linearly independent in a vector space, then the dimension of the space is at least n. The pivot columns of a matrix form a basis for its column space.
Interesting Theory Let V be a finite-dimensional vector space with Dim(V ) = n and S V. If S is independent and S < n, then it can be expanded to create a basis. If S spans V and S > n, then vectors can be removed in such a way that we eventually obtain a basis.
Homework Pages: 226-228 12, 20, 26 Pages: 242-244 1, 4, 6, 7, 11, 15, 19, 35, 36, 47
Maps, Functions, and Transformations Definition We say f is a map, mapping, function, or transformation from set X to set Y, written f : X Y, if each x X is associated a unique element f (x) Y. The element f (x) is the image of x. The set X is the domain of f. The set Y is the co-domain of f. The range or image of f, is the set Range(f ) = Image(f ) = {f (x) x X }. Note: Range(f ) Y and need not equal Y!!
One-To-One, Onto, and Bijections Definition Let f : X Y be a function. f is injective or one-to-one if it maps distinct entities in the domain to distinct entities in the range. That is, f (a) = f (b) = a = b. f is surjective or onto if its range (image) and co-domain are the same. That is, y Y, x X such that f (x) = y. f is a bijection if it is one-to-one and onto.
Examples Let s Do It! Complete the table for a function f : X Y which is given by f (x) = x 2. Domain Co-Domain Range 1-1 Onto Bijection R R (, 0] R (, 0] [0, ) (2, 3) [4, 9] [ 3, 3] [0, 9]
Transformations Definition Suppose V and W are vector spaces with common scalar set F. Let f : V W be a function from V to W. We say f is a linear transformation if f (αx + βy) = αf (x) + βf (y) for all vectors x, y V and all scalars α, β F. If f is also a bijection (1-1 and onto), then we say that f is an isomorphism and the spaces X and Y are isomorphic to each other.
Transformation Example Let s Do It! [ ] 1 2 3 Suppose A =. Define f : R 0 2 7 3 R 2 by f (x) = Ax. Show that this is a linear transformation.
Transformation Theory Let A be a m n matrix. The mapping x Ax is a linear transformation from R n to R m.
Example Example Define f : R 3 R 2 by f x 1 x 2 x 3 [ x1 + 2x = 2 + 3x 3 2x 2 7x 3 Show that this is a linear transformation by using the previous theorem, not the definition! ].
Transformations from R n to R m Let T be a linear transformation from R n to R m. Then there is an m n matrix A such that T (x) = Ax for all x R n. The matrix A is called the transformation matrix.
Transformation Theory A linear transformation from R n to R m is completely determined by the images of the standard unit vectors in R n, which we have written as e 1, e 2,..., e n. Corollary The transformation matrix, A, of a linear transformation T : R n R m is given by A = [T (e 1 ) T (e 2 )... T (e n )].
Using Transformations Example Suppose that T : R 3 R 2 is a linear transformation and 1 [ ] 1 [ ] T 2 1 =, T 0 2 =, and 2 5 3 1 2 [ ] T 2 4 =. Find the transformation matrix, A, of 3 1 the linear transformation T. Note: 1 1 2 1 0 0 1 0 0 1 3 2 0 2 0 1 0 rref 2 1 5 0 1 0 2 2 1 3 1 1 0 0 1 0 0 1 1 2 1
When are Transformations Onto? Let A be an m n matrix. For the linear map x Ax to be surjective (onto), it is necessary and sufficient that the columns of A span R m.
When are Transformations 1-1? Definition Let V be a vector space. The kernel of a linear map T : V W is Ker(T ) = {x V T (x) = 0}. Let V be a vector space and T : V W a linear map. The kernel of a linear transformation is a subspace of V. In order that the linear map T (x) = Ax be injective (1-1), it is necessary and sufficient that the kernel of T contain only the 0-vector.
Transformations - Kernel and Image Definition If f is a mapping of a set X to a set Y and U X, then the image of U is f [U] = {f (x) x U}. Definition Let f : X Y be a mapping between sets and suppose S Y. Then the inverse image of S is f 1 [S] = {x X f (x) S}.
Transformations - Kernel, Range, Images, and Inverse Images of Let T : V W be a linear transformation between vector spaces. Then The kernel (null space) of T is a subspace of V. The range (image of V ) of T is a subspace of W. Let X be a subspace of V. Then T [X ] is a subspace of W. Let Y be a subspace of W. Then T 1 [Y ] is a subspace of V.
Example Example Consider the linear map of x = (x 1, x 2 ) R 2 defined by T (x) = x 1 + 2x 2. What is the image of the map? What is the co-domain? What is the matrix associated with this map? Graph the kernel of the map.
Example Example 1 2 3 Let A = 2 3 1 Define a linear transformation by 3 4 5 T (x) = Ax. Describe the kernel and range in simple terms.
Example of Kernel and Range Example 23 7 5 2 Let A =. Find the kernel and range of the 7 5 0 2 linear transformation.
Coordinate Vectors Definition Let B = {u 1,..., u n } be an ordered basis. Suppose that x = α 1 u 1 + + α n u n = n α i u i. Then, the coordinate vector of x associated with the ordered basis B is α 1 [x] B =. α n. i=1
Example Example Let p 1 (t) = t 2 + 4, p 2 (t) = t 1, p 3 (t) = t 3 + t 2, and p 4 (t) = t 2 + 2t + 2. From a previous example, we know that an ordered basis for the Span({p 1, p 2, p 3, p 4 }) is Find [p 4 ] B. B = {p 1, p 2, p 3 }.
Equivalence Relations Definition A relation (written as ) is an equivalence relation on a set X if it has three properties for all x, y, z X : 1. Reflexive: x x 2. Symmetric: If x y, then y x 3. Transitive: If x y and y z, then x z. The relation of isomorphism between two vector spaces is an equivalence relation.
Coordinate Vector as a Mapping Let B be a basis for a vector space V and x V. Define f (x) = [x] B. Then f (x) is a linear transformations. Then f (x) is injective (one-to-one). Then f (x) is surjective (onto). That is, f is an isomorphism. Corollary Let V be an n-dimensional vector space. Then, using the transformation above, we have that V and R n are isomorphic. Moreover, all n-dimensional vector spaces are isomorphic.
When are Two Finite-al s Isomorphic? Let B = {v 1,..., v n } be a basis for a vector space V. Suppose L : V W is an isomorphism. Then {L(v 1 ),..., L(v n )} is a basis for W. Corollary Two vector spaces are isomorphic if and only if their dimensions agree. Example The vector spaces R n+1 and P n are isomorphic to each other.
Change of Coordinates Recall x [x] B is a linear transformation. Let V be a n-dimensional vector space in which two ordered bases B = {u 1,..., u n } and C = {v 1,..., v n } have been prescribed. Let P be the transition matrix whose columns are the C-coordinates of the basis vectors in B. That is, let Then for all x V, we have P = [[u 1 ] C [u 2 ] C [u n ] C ] [x] C = P [x] B. Sometimes P is denoted by P C B.
Example Example Suppose B = {x 2 + 2x + 3, 3x 2 2, 5x 2 2x + 1} and C = {2x 2 + x, 2x 2 + x + 2, x + 3} are ordered bases for P 2. 1. Find the transition matrix, P, needed for converting B-coordinates into C-coordinates. 2. Find the transition matrix, P, needed for converting C-coordinates into B-coordinates.
Big Boy A linear transformation from one vector space to another is completely determined by the images of any basis in the domain. These images can, in turn, be completely arbitrary.
More Theory n If a i u i = 0 and if L is linear, then i=1 n a i L(u i ) = 0. Let L be a linear transformation from a finite-dimensional vector space X into a vector space Y. Let B = {u 1,..., u n } and C be ordered bases for X and L(X ), respectively. Let A be the matrix whose columns are the vectors [L(u i )] C. Then for all x X, the equation [L(x)] C = A [x] B i=1 is valid. Moreover, the matrix A is unique and is called the representation of L with respect to the ordered bases B and C.
Example Example Let L : M 2 2 R 3 be given by ([ a b L c d ]) = 2a 3b c + d 4a + b + 2d. Use the standard basis for M 2 2, that is B = Let C = {[ 1 0 0 0 1 2 3, ] [ 0 1, 0 0 0 2 1, ] [ 0 0, 1 0 2 1 1 ] [ 0 0, 0 1 ]}. be a basis for R3. Find A such that [L(M)] C = A [M] B for all M M 2 2.
Similar Matrices Definition Let A and B be two n n matrices. We say that A is similar to B if there is a invertible matrix P such that A = PBP 1. We write A B. Similarity is an equivalence relation. That is, 1. Reflexive: A A. 2. Symmetric: If A B, then B A. 3. Transitive: If A B and B C, then A C. This is a homework problem.
Homework Pages: 242-244 16, 42 Pages: 267-269 1, 3, 6, 7, 13, 15, 32, 33, 37 Pages: 397-399 1 Pages: 413-414 1, 6
Row and Column of a Matrix Definition Let A be an n n matrix. The row space of A is the span of the rows of the matrix A. The column space of A is the span of the columns of the matrix A.
Equivalent Matrices If A and B are two row (column) equivalent matrices, then the row (column) spaces of A and B are equal.
Example Example Consider the linear transformation from R 4 to R 3 defined by T (x) = Ax where 1 2 7 3 A = 2 1 4 0. 3 4 17 2 Find a simple basis for the row space of the matrix A. Find a simple basis for the column space of the matrix A. (Why must we use columns from the original matrix, A?) Find a simple basis for the null space of A. Find a simple basis for the image of A. (Yup, we ve already calculated this one!)
Row and Column s of a Matrix The dimension of the row space of a matrix equals the dimension of the column space of a matrix.
Rank and Nullity of a Matrix Definition The dimension of the row space of a matrix is called the row rank of the matrix. The dimension of the column space of a matrix is called the column rank of the matrix. Since the dimensions of the row space of a matrix equals that of the column space, we can talk about the rank of a matrix, which equal the row (or column) rank of the matrix. The dimension of the null space of a matrix is called the nullity of the matrix.
Relationship Between and Range of a Transformation If T is a linear transformation whose domain is an n-dimensional vector space, then Dim[Ker(T )] + Dim[Range(T )] = n. [ ] Let A be an m n matrix. The number of columns, n, equals the dimension of the column space plus the dimension of the null space. That is, Dim[Null(A)] + Dim[Col(A)] = n. That is, Rank(A) + Nullity(A) = n.
Some Equivalent Statements Let A be an n n matrix. The following are equivalent: Rank(A) = n A is row equivalent to I n. A has an inverse. det(a) 0. The homogeneous system Ax = 0 has only the trivial solution. The system Ax = b has a unique solution. The nullity of A is zero. The rows of A are linearly independent. The columns of A are linearly independent.
Homework Pages: 267-269 29, 36 Pages: 282-285 2, 5, 10, 12, 16, 20, 22, 34, 37, 50 In Mathematica, the command MatrixRank[A] which will give the rank of the matrix A.