Page 32 - Vector Analysis

28 CHAPTER 1. Linear Algebra

1.7 Representation of Linear Transformations

In Section 1.6.1, we see that any m ˆ n matrix is associated with a linear map. On the

other hand, suppose that V is a n-dimensional vector space with basis B = ...t¨v¨j¨u...njv=n1], and
W is [ and
W= a m-dimensional vector space with basis Br = twiuim=1. Define V = v1
[ ] P L (V, W). Since Lvj P W, for each 1 ď j ďn
w1 ... ¨ ¨ ¨ ... wm , and let L we can write

m

Lvj = ř aijwi for some coefficients aij. Moreover, if u P V, then

i=1

n

u = ÿ cjvj or c = [u]B or u = Vc ,

j=1

and by the linearity of L,

Lu ( n ) n cj Lvj n m cj aij wi m( n )
L cj vj ÿ aijcj wi
ÿ ÿ ÿ ÿ ÿ

= = = = .

j=1 j=1 j=1 i=1 i=1 j=1

n

Let bi = ř aijcj, and b = [b1, ¨ ¨ ¨ , bm]T. Then

j=1

[Lu]Br = b = Ac = A[u]B .

The discussion above induces the following

Definition 1.91. Let V, W be two vector spaces, dim(V) = n and dim(W) = m, and
B, Br are basis of V, W, respectively. For L P L (V, W), the matrix representation of L
relative to bases B and Br, denoted by [L]Br,B, is the matrix satisfying

[Lu]Br = [L]Br,B[u]B @ u P V .

If L P L (V, V), we simply use [L]B to denote [L]B,B.

1.8 Matrix Diagonalization

Definition 1.92 (Eigenvalues and Eigenvectors). Let V be a finite dimensional vector spaces
over a scalar field F, and L P B(V). A scalar λ P F is said to be an eigenvalue of L if
there is a non-zero vector v P V such that Lv = λv. The collection of all eigenvalues of L
is denoted by σ(L).

For an eigenvalue λ P F of L, a non-zero vector v P V satisfying Lv = λv is called an
eigenvector associated with the eigenvalue λ, and the collection of all v P V such that
Lv = λv is called the eigenspace associated with λ.