Linear Maps and Matrices

Transcription

1 Linear Maps and Matrices Maps Suppose that V and W are sets A map F : V W is a function; that is, to every v V there is assigned a unique element w F v in W Two maps F : V W and G : V W are equal if F v Gv for all v V An important operation on maps is composition, which plays a major role in linear maps Suppose that G : W X is a map The composition G F : V X is defined by for v V Linear maps G F v GF v Definition Suppose that V and W are vector spaces A map L : V W is linear if: Suppose that v, v V Then Suppose that v V and c R Then L v + v L v + L v Lc v cl v Theorem Suppose that L : V W is a linear map Then L V W, L v L v for v V, Suppose that v,, v n V and c,, c n R Then Lc v + + c n v n c L v + c L v + + c n L v n Example Determine if the map L : R R defined by x x + x L x x + 6x for is linear x x R Solution: L is linear Suppose that v y + z Lv + v L y + z y y and v z z R y + z + y + z y + z + 6y + z y + y z + z + Lv y + 6y z + 6z + Lv y + y + z + z y + 6y + z + 6z

2 Suppose that Thus L is a linear map v cy Lcv L cy cy + y cy + 6y y y R and c R cy + cy cy + 6cy c y + y clv y + 6y To show that a map is not linear, we must give specific vectors or scalars which give a counterexample to or of the definition of a linear map Example Determine if L : R R defined by Lx, x x, x for x, x R is a linear map Solution: L is not linear Let v, and c Then which is not equal to clv L,,, Lcv L,,, Thus of the definition of a linear map does not hold for L Recall that C i [a, b] is the vector space of i times continuously differentiable functions on the closed interval [a, b] By the rules of calculus, the map S : C i [a, b] R defined by Sf b a fdx for f C i [a, b] is a linear map and D : C i+ [a, b] C i [a, b] defined by is a linear map Df df dx for f Ci+ [a, b] Suppose that L : V W is a linear map The Kernel of L is The Image of L is KernelL { v V L v W } ImageL {L v v V } KernelL is a subspace of V and ImageL is a subspace of W Linear maps from R n to R m Suppose that A R m n is an m n matrix Then we have a linear map L A : R n R m, defined by L A x A x for x R n In fact, the maps of this type are the only linear maps from R n to R m

3 Example Let A Then the linear map L A : R R satisfies L A x x A x x x x for x x x x x x R x + x + x x + x + x Theorem 6 Suppose that L : R n R m is a linear map Then there is a unique m n matrix A such that L L A Proof We will prove the existence of such a matrix A Set For x x,, x n T R n, we have A Le, Le,, Le n L x Lx e + x e + + x n e n x Le + x Le + + x n Le n since L is linear, Le, Le,, Le n x A x L A x L L A since L x L A x for all x R n Definition 7 Suppose that L : R n R m is a linear map The m n matrix A Le,, Le n which has the property that L L A is called the matrix of L Example 8 Let L : R R be defined by L x x x + x + x x x + x + x for The matrix of L is A L, L x x x, L R Example verifies the calculation of Example 8 It is not difficult to compute bases of the Kernel and Image of a linear map from R n into R m, since we have the following identities:

4 KernelL A NA and ImageL A Column SpaceA Example 9 Find bases of the kernel and image of the linear map L : R R defined by x x + x L x x + 6x for x x R Compute the dimension of the kernel of L and the image of L Solution: The matrix of L is A Le, Le We compute that the RRE form of A is Since the only leading one is in the first column, { } 6 is a basis of ImageL Column SpaceA by the Algorithm to refine the columns of a matrix to a basis of its column space of Lecture Note 6, and dim ImageL To find a basis of NA, we write x t x t with t R Thus { t is a basis of NA KernelL by the Algorithm to compute a basis of a null space of a matrix of Lecture Note 6, and dim KernelL Suppose that L A : R n R m and L B : R m R l Then the composition L B L A : R n R l, defined by L B L A v L B L A v for v R n satisfies } L B L A L BA where BA is the l n matrix obtained by multiplying the l m matrix B and the m n matrix A Distance preserving linear maps

5 Example Given an angle θ, Let cosθ sinθ R θ sinθ cosθ L Rθ : R R is the linear map which rotates a vector v R counterclockwise about the origin by the angle θ Example Let A L A : R R is the linear map which reflects a vector v R about the x-axis Recall that the length of a vector is v x + x v x x R Theorem Page 7 of M Artin s Algebra, first edition The linear maps L : R R which preserve length L v v for all v R are the rotations Example and reflections about a line through the origin Volume Suppose that v,, v n R n Let P {t v + + t n v n t,, t n R and t i for i n} P is the box spanned by v,, v n Suppose that L L A : R n R n is a linear map Then LP is the box spanned by Lv,, Lv n We compare the volumes of P and LP by the formula vollp DetA volp To prove this, we use the formula for computing the volume of P using a determinant from Lecture Note, the fact that determinant preserves matrix multiplication and the formulas for matrix multiplication from the beginning of Lecture Note to calculate vollp DetLv, Lv,, Lv n DetAv, Av,, Av n DetAv,, v n DetADetv,, v n DetA volp If we further assume that L L A is distance preserving, then we can calculate that DetA ±, so that vollp volp Inverses of linear maps A map F : S T is - if for all p, q S, F p F q implies p q If F is linear, then F is - if and only if KernelF { S }

6 If F is -, then S is the only element which can map to T Suppose that KernelF { S }, and p, q S are such that F p F q Then F p q F p F q W so p q KernelF, and p q V Thus p q A map F : S T is onto if for all v T, there exists q S such that F p q If F is linear, then F is onto if and only if ImageF T In Example 9, we computed that ImageL R so L is not onto and KernelL { } so L is not - A map F : S T is invertible if there exists a map G : T S such that G F is the identity map id S on S and F G is the identity map id T on T ; that is, GF x x for all x S and F Gy y for all y T A map F is invertible if and only if F is - and onto If F is invertible then the map G is unique We will write G F and say that G is the inverse of F Theorem Suppose that L : V W is a linear map which is - and onto Then the inverse map L : W V is a linear map A - and onto linear map L : V W is called an isomorphism Theorem Suppose that A is an m n matrix Then the linear map L A : R n R m is an isomorphism if and only if n m and DetA If L A is an isomorphism, then the inverse map of L A is L A L A, where A is the inverse matrix of A Vector spaces V and W are called isomorphic if there exists an isomorphism L : V W Isomorphic vector spaces are indistinguishable as vector spaces Theorem Suppose that V is an n-dimensional vector space Then V is isomorphic to R n Let {v,, v n } be a basis of V Theorem is proven by showing that the map ϕ : V R n defined by ϕ v v is an isomorphism of vector spaces The inverse map ϕ : R n V is ϕ x x n x v + + x n v n These isomorphisms are useful in understanding the algorithm of the next section The matrix of a linear map with respect to a given basis Suppose that F : V W is a linear map Let {v,, v n } be a basis of V, and {w,, w m } be a basis of W The matrix M F of the linear map F with respect to the bases of V and of W is the unique m n matrix M F which has the property that M F v F v 6

7 for all v V It follows that M F F v, F v,, F v n The matrix of a linear map L : R n R m, defined after Theorem 6 as A Le,, Le n, is then the the matrix of L with respect to the standard bases of R n and R m The relationship between F and L M F is shown in the following diagram; the vertical arrows are isomorphisms identifying V with R n and W with R m respectively V ϕ R n F L M F W ϕ R m Often it is best to compute M F directly from This works particularly well when is a basis such as a standard basis in which it is easy to compute coordinate vectors The computation of M F directly from is done by solving for the coefficients of the matrix A a ij in the system of equations to get F v a w + a w + + a m w m F v a w + a w + + a m w m F v n a n w + a n w + + a nm w m M F A T After finding the matrix of a linear map F, bases of the Kernel of F and Image of F can be computed from formulas and Example 6 Let U span, cosx, sinx C, where C is the vector space of infinitely differentiable functions, and let D : U U be differentiation, cosx, sinx are linearly independent by the Wronskian criterion of Lecture Note, and so {, cosx, sinx} is a basis of U Compute the matrix M D using equation Find a basis of the Kernel of D Find a basis of the Image of D Solution: We have D + cosx + sinx Dcosx sinx + cosx + sinx Dsinx cosx + cosx + sinx 7

8 Thus M D D, Dcos x, Dsin x The RRE form of M D is Writing with t R, we see that x x x t t is a basis of KernelL M D NM D by the algorithm to compute a basis of a null space of Lecture Note 6 Thus {} is a basis of the kernel of D, since Since the second and third columns of the RRE form of M D have leading ones, a basis of Column SpaceM D ImageL M D is, by the Algorithm to refine the columns of a matrix to a basis of its column space of Lecture Note 6 so a basis of the image of D is since sinx { sinx, cosx} and cosx If F is the identity map id so that V W, then M id is the transition matrix M defined in Lecture Note 6 Suppose that G : W X is a linear map, and is a basis of X We have a linear map G F : V X We have the important formula 8

9 6 M G F M GM F A particularly important application of this formula is 7 M F M id V M F M id V S M F S, where F : V V is linear, and are bases of V, S M and S M Definition 7 Suppose that A and B are n n matrices B is similar to A if there exists an invertible n n matrix S such that B S AS We have the following result which generalizes the criterion of Theorem Theorem 8 Suppose that L : V W is a linear map of finite dimensional vector spaces, is a basis of V and is a basis of W Then L is invertible if and only if dim V dim W and DetM L If L is invertible, then M L M L Theorem 9 Suppose that V and W are finite dimensional vector spaces and L : V W is a linear map Then dimkernell + dimimagel dimv Proof Let n be the dimension of V and m be the dimension of W Let be a basis of V and be a basis of W Let A M L be the matrix of L with respect to the bases and A is an m n matrix We have that the nullity of A is the dimension of the Kernel of L Taking the coordinate vector with respect to is an isomorphism of the Kernel of L and the null space NA and the rank of A is the dimension of the Image of L Taking the coordinate vector with respect to is an isomorphism of the Image of L and the column space of A Now by the rank nullity theorem we have that dimv n ranka + nullitya dimimagel + dimkernell An Algorithm to compute the matrix of a linear map with respect to given bases Let F : V W be a linear map, and suppose that {v,, v n } is a basis of V and {w,, w m } is a basis of W Let be a basis of W which is easy to compute with such as a standard basis of W The m m + n matrix w, w,, w m F v,, F v n is transformed by elementary row operations into the reduced row echelon form I m M F Example Let L L A : R R be the linear map where A 9

10 Find the matrix M L where { v, v } and { w, w } Use your answer to part to compute L v if v Solution: We compute Lv L and We form the matrix Lv L w w Lv Lv which has the RRE form Thus M L solving part To solve part we compute L v M L v, Example Let L : P P be the linear map defined by Lc + c x c + c + c + c x for c + c x P Compute M L where {v + x, v + x} and {w + x, w x} Solution: The standard basis of P is {, x} We compute and Lv L + x x + 6x taking c, c Lv L + x x + 8x taking c, c We form the matrix w w Lv Lv 6 8

11 which has the RRE form so M L