Chapter 2 – Linear Transformations and Matrices ... - Per-Olof Persson

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Chapter 2 – Linear Transformations and Matrices. Per-Olof Persson persson@ berkeley.edu. Department of Mathematics. University of California, Berkeley.
Linear Transformations

Chapter 2 – Linear Transformations and Matrices

Definition We call a function T : V → W a linear transformation from V to W if, for all x, y ∈ V and c ∈ F , we have (a) T(x + y) = T(x) + T(y) and

(b) T(cx) = cT(x) Per-Olof Persson [email protected] Department of Mathematics University of California, Berkeley

1

If T is linear, then T(0 ) = 0

2

T is linear ⇐⇒ T(cx + y) = cT(x) + T(y) ∀x, y ∈ V, c ∈ F

3 4

Math 110 Linear Algebra

If T is linear, then T(x − y) = T(x) − T(y) ∀x, y ∈ V

T isPlinear ⇐⇒ for Px1 , . . . , xn ∈ V and a1 , . . . , an ∈ F , T ( ni=1 ai xi ) = ni=1 ai T(xi )

Special linear transformations

The identity transformation IV : V → V: IV (x) = x, ∀x ∈ V The zero transformation T0 : V → W: T0 (x) = 0 ∀x ∈ V

Null Space and Range Definition For linear T : V → W, the null space (or kernel) N(T) of T is the set of all x ∈ V such that T(x) = 0 : N(T) = {x ∈ V : T(x) = 0 } The range (or image) R(T) of T is the subset of W consisting of all images of vectors in V: R(T) = {T(x) : x ∈ V} Theorem 2.1 For vector spaces V, W and linear T : V → W, N(T) and R(T) are subspaces of V and W, respectively. Theorem 2.2 For vector spaces V, W and linear T : V → W, if β = {v1 , . . . , vn } is a basis for V, then

Nullity and Rank

Definition For vector spaces V, W and linear T : V → W, if N(T) and R(T) are finite-dimensional, the nullity and the rank of T are the dimensions of N(T) and R(T), respectively. Theorem 2.3 (Dimension Theorem) For vector spaces V, W and linear T : V → W, if V is finite-dimensional then nullity(T) + rank(T) = dim(V)

R(T) = span(T(β)) = span({T(v1 ), . . . , T(vn )})

Properties of Linear Transformations

Theorem 2.4 For vector spaces V, W and linear T : V → W, T is one-to-one if and only if N(T) = {0 }. Theorem 2.5 For vector spaces V, W of equal (finite) dimension and linear T : V → W, the following are equivalent: (a) T is one-to-one

(b) T is onto (c) rank(T) = dim(V)

Linear Transformations and Bases

Theorem 2.6 Let V, W be vector spaces over F and {v1 , . . . , vn } a basis for V. For w1 , . . . , wn in W, there exist exactly one linear transformation T : V → W such that T(vi ) = wi for = 1, . . . , n. Corollary Suppose {v1 , . . . , vn } is a finite basis for V, then if U, T : V → W are linear and U(vi ) = T(vi ) for i = 1, . . . , n, then U = T.

Coordinate Vectors

Matrix Representations

Definition For a finite-dimensional vector space V, an ordered basis for V is a basis for V with a specific order. In other words, it is a finite sequence of linearly independent vectors in V that generates V. Definition Let β = {u1 , . . . , un } be an ordered basis for V, and for x ∈ V let a1 , . . . , an be the unique scalars such that n X x= ai ui . i=1

The coordinate vector of x relative to β is   a1   [x]β =  ...  an

Addition and Scalar Multiplication Definition Let T, U : V → W be arbitrary functions of vector spaces V, W over F . Then T + U, aT : V → W are defined by (T + U)(x) = T(x) + U(x) and (aT)(x) = aT(x), respectively, for all x ∈ V and a ∈ F . Theorem 2.7 With the operations defined above, for vector spaces V, W over F and linear T, U : V → W: (a) aT + U is linear for all a ∈ F

Definition Suppose V, W are finite-dimensional vector spaces with ordered bases β = {v1 , . . . , vn }, γ = {w1 , . . . , wm }. For linear T : V → W, there are unique scalars aij ∈ F such that T(vj ) =

m X

for 1 ≤ j ≤ n.

aij wi

i=1

The m × n matrix A defined by Aij = aij is the matrix representation of T in the ordered bases β and γ, written A = [T]γβ . If V = W and β = γ, then A = [T]β . Note that the jth column of A is [T(vj )]γ , and if [U]γβ = [T]γβ for linear U : V → W, then U = T.

Matrix Representations

Theorem 2.8 For finite-dimensional vector spaces V, W with ordered bases β, γ, and linear transformations T, U : V → W: (a) [T + U]γβ = [T]γβ + [U]γβ

(b) [aT]γβ = a[T]γβ for all scalars a

(b) The collection of all linear transformations from V to W is a vector space over F Definition For vector spaces V, W over F , the vector space of all linear transformations from V into W is denoted by L(V, W), or just L(V) if V = W.

Composition of Linear Transformations

Theorem 2.9 Let V, W, Z be vector spaces over a field F , and T : V → W, U : W → Z be linear. Then UT : V → Z is linear. Theorem 2.10 Let V be a vector space and T, U1 , U2 ∈ L(V). Then

(a) T(U1 + U2 ) = TU1 + TU2 and (U1 + U2 )T = U1 T + U2 T

(b) T(U1 U2 ) = (TU1 )U2 (c) TI = IT = T (d) a(U1 U2 ) = (aU1 )U2 = U1 (aU2 ) for all scalars a

Matrix Multiplication Let T : V → W, U : W → Z, be linear, α = {v1 , . . . , vn }, β = {w1 , . . . , wm }, γ = {z1 , . . . , zp } ordered bases for U, W, Z, and A = [U ]γβ , B = [T ]βα . Consider [UT]γα : ! m m X X (UT)(vj ) = U(T(vj )) = U Bkj wk = Bkj U(wk ) k=1

=

m X

Bkj

k=1

p X i=1

Aik zi

!

k=1

=

p m X X i=1

k=1

Aik Bkj

!

zi

Definition Let A, B be m × n, n × p matrices. The product AB is the m × p matrix with n X (AB)ij = Aik Bkj , for 1 ≤ i ≤ m, 1 ≤ j ≤ p k=1

Matrix Multiplication

Properties

Theorem 2.11 Let V, W, Z be finite-dimensional vector spaces with ordered bases α, β, γ, and T : V → W, U : W → Z be linear. Then [UT]γα =

[U]γβ [T]βα

Corollary Let V be a finite-dimensional vector space with ordered basis β, and T, U ∈ L(V). Then [UT]β = [U]β [T]β . Definition The Kronecker delta is defined by δij = 1 if i = j and δij = 0 if i 6= j. The n × n identity matrix In is defined by (In )ij = δij .

Theorem 2.12 Let A be m × n matrix, B, C be n × p matrices, and D, E be q × m matrices. Then (a) A(B + C) = AB + AC and (D + E)A = DA + EA

(b) a(AB) = (aA)B = A(aB) for any scalar a (c) Im A = A = AIn (d) If V is an n-dimensional vector space with ordered basis β, then [IV ]β = In Corollary Let A be m × n matrix, B1 , . . . , Bk be n × p matrices, C1 , . . . , Ck be q × m matrices, and a1 , . . . , ak be scalars. Then ! ! k k k k X X X X A ai Bi = ai ABi and a i Ci A = ai Ci A i=1

Properties

i=1

i=1

i=1

Left-multiplication Transformations

Theorem 2.13 Let A be m × n matrix and B be n × p matrix, and uj , vj the jth columns of AB, B. Then (a) uj = Avj (b) vj = Bej Theorem 2.14 Let V, W be finite-dimensional vector spaces with ordered bases β, γ, and T : V → W be linear. Then for u ∈ V: [T(u)]γ =

[T]γβ [u]β

Definition Let A be m × n matrix. The left-multiplication transformation LA is the mapping LA : Fn → Fm defined by LA (x) = Ax for each column vector x ∈ Fn . Theorem 2.15 Let A be m × n matrix, then LA : Fn → Fm is linear, and if B is m × n matrix and β, γ are standard ordered bases for Fn , Fm , then: (a) [LA ]γβ = A (b) LA = LB if and only if A = B (c) LA+B = LA + LB and LaA = aLA for all a ∈ F

(d) For linear T : Fn → Fm , there exists a unique m × n matrix C such that T = LC , and C = [T]γβ (e) If E is an n × p matrix, then LAE = LA LE (f) If m = n then LIn = IFn

Associativity of Matrix Multiplication

Inverse of Linear Transformations Definition Let V, W be vector spaces and T : V → W be linear. A function U : W → V is an inverse of T if TU=IW and UT=IV . If T has an inverse, it is invertible and the inverse T−1 is unique.

Theorem 2.16 Let A, B, C be matrices such that A(BC) is defined. Then (AB)C is also defined and A(BC) = (AB)C.

For invertible T,U: 1

(TU)−1 = U−1 T−1

2

(T−1 )−1 = T (so T−1 is invertible)

3

If V,W have equal dimensions, linear T : V → W is invertible if and only if rank(T) = dim(V)

Theorem 2.17 For vector spaces V,W and linear and invertible T : V → W, T−1 : W → V is linear.

Inverses

Definition An n × n matrix A is invertible if there exists an n × n matrix B such that AB = BA = I. Lemma For invertible and linear T from V to W, V is finite-dimensional if and only if W is finite-dimensional. Then dim(V) = dim(W). Theorem 2.18 Let V,W be finite-dimensional vector spaces with ordered bases β, γ, and T : V → W be linear. Then T is invertible if and only if [T]γβ is invertible, and [T−1 ]βγ = ([T]γβ )−1 .

Isomorphisms

Definition Let V,W be vector spaces. V is isomorphic to W if there exists a linear transformation T : V → W that is invertible. Such a T is an isomorphism from V onto W. Theorem 2.19 For finite-dimensional vector spaces V,W, V is isomorphic to W if and only if dim(V) = dim(W). Corollary A vector space V over F is isomorphic to Fn if and only if dim(V) = n.

The Standard Representation

Definition Let β be an ordered basis for an n-dimensional vector space V over the field F . The standard representation of V with respect to β is the function φβ : V → Fn defined by φβ (x) = [x]β for each x ∈ V. Theorem 2.21 For any finite-dimensional vector space V with ordered basis β, φβ is an isomorphism.

Inverses

Corollary 1 For finite-dimensional vector space V with ordered basis β and linear T : V → V, T is invertible if and only if [T]β is invertible, and [T−1 ]β = ([Tβ ])−1 . Corollary 2 An n × n matrix A is invertible if and only if LA is invertible, and (LA )−1 = LA−1 .

Linear Transformations and Matrices

Theorem 2.20 Let V,W be finite-dimensional vector spaces over F of dimensions n, m with ordered bases β, γ. Then the function Φ : L(V, W) → Mm×n (F ), defined by Φ(T) = [T]γβ for T ∈ L(V, W), is an isomorphism. Corollary For finite-dimensional vector spaces V,W of dimensions n, m, L(V, W) is finite-dimensional of dimension mn.

The Change of Coordinate Matrix

Theorem 2.22 Let β and β 0 be ordered bases for a finite-dimensional vector space V, and let Q = [IV ]ββ 0 . Then (a) Q is invertible (b) For any v ∈ V, [v]β = Q[v]β 0 Q = [IV ]ββ 0 is called a change of coordinate matrix, and we say that Q changes β 0 -coordinates into β-coordinates. Note that if Q changes from β 0 into β coordinates, then Q−1 changes from β into β 0 coordinates.

Linear Operators

Linear Functionals

A linear operator is a linear transformation from a vector space V into itself. Theorem Let T be a linear operator on a finite-dimensional vector space V with ordered bases β, β 0 . If Q is the change of coordinate matrix from β 0 into β-coordinates, then [T]β 0 = Q−1 [T]β Q Corollary Let A ∈ Mn×n (F ), and γ an ordered basis for Fn . Then [LA ]γ = Q−1 AQ, where Q is the n × n matrix with the vectors in γ as column vectors. Definition For A, B ∈ Mn×n (F ), B is similar to A if the exists an invertible matrix Q such that B = Q−1 AQ.

Coordinate Functions

A linear functional on a vector space V is a linear transformation from V into its field of scalars F . Example Let V be the continuous real-valued functions on [0, 2π]. For a fix g ∈ V, a linear functional h : V → R is given by Z 2π 1 h(x) = x(t)g(t) dt 2π 0 Example Let V = Mn×n (F ), then f : V → F with f(A) = tr(A) is a linear functional.

Dual Spaces

Example Let β = {x1 , . . . , xn } be a basis for a finite-dimensional vector space V. Define fi (x) = ai , where   a1   [x]β =  ...  an

is the coordinate vector of x relative to β. Then fi is a linear functional on V called the ith coordinate function with respect to the basis β. Note that fi (xj ) = δij .

Dual Bases

Definition For a vector space V over F , the dual space of V is the vector space V∗ = L(V, F ). Note that for finite-dimensional V, dim(V∗ ) = dim(L(V, F )) = dim(V) · dim(F ) = dim(V) so V and V∗ are isomorphic. Also, the double dual V∗∗ of V is the dual of V∗ .

Double Dual Isomorphism

Theorem 2.24 Let β = {x1 , . . . , xn } be an ordered basis for finite-dimensional vector space V, and let fi be the ith coordinate function w.r.t. β, and β ∗ = {f1 , . . . , fn }. Then β ∗ is an ordered basis for V∗ and for n any f ∈ V∗ , X f= f(xi )fi . i=1

Definition The ordered basis β ∗ = {f1 , . . . , fn } of V∗ that satisfies fi (xj ) = δij is called the dual basis of β. Theorem 2.25 Let V, W be finite-dimensional vector spaces over F with ordered bases β, γ. For any linear T : V → W, the mapping Tt : W∗ → V∗ defined by Tt (g) = gT for all g ∈ W∗ is linear with the property ∗ [Tt ]βγ ∗ = ([Tγβ )t .

For a vector x ∈ V, define x ˆ : V∗ → F by x ˆ(f) = f(x) for every ∗ f ∈ V . Note that x ˆ is a linear functional on V∗ , so x ˆ ∈ V∗∗ . Lemma For finite-dimensional vector space V and x ∈ V, if x ˆ(f) = 0 for all f ∈ V∗ , then x = 0. Theorem 2.26 Let V be a finite-dimensional vector space, and define ψ : V → V∗∗ by ψ(x) = x ˆ. Then ψ is an isomorphism. Corollary For finite-dimensional V with dual space V∗ , every ordered basis for V∗ is the dual basis for some basis for V.