Kunze Linear Algebra · Problem 3.4.13

Linear Transformations

Mathematics — Methods · matrix-of-transformation-basis-for-operators

Problem

Let

V

and

W

be finite-dimensional vector spaces over the field

F

and let

T

be a linear transformation from

V

into

W

. If

\mathfrak{B} = \{\alpha_1, \ldots, \alpha_n\} \quad \text{and} \quad \mathfrak{B}' = \{\beta_1, \ldots, \beta_m\}

are ordered bases for

V

and

W

, respectively, define the linear transformations

E^{p,q}

as in the proof of Theorem 5:

E^{p,q}(\alpha_i) = \delta_{iq}\beta_p

. Then the

E^{p,q}

1 \leq p \leq m

1 \leq q \leq n

, form a basis for

L(V, W)

, and so

T = \sum_{p=1}^{m} \sum_{q=1}^{n} A_{pq} E^{p,q}

for certain scalars

A_{pq}

(the coordinates of

T

in this basis for

L(V, W)

). Show that the matrix

A

with entries

A(p, q) = A_{pq}

is precisely the matrix of

T

relative to the pair

\mathfrak{B}

\mathfrak{B}'

Intuitive explanation

The key insight is that the coordinates of $T$ in the basis $\{E^{p,q}\}$ for $L(V,W)$ encode exactly where $T$ sends basis vectors of $V$ . Each $E^{p,q}$ is a "selector" that picks out one basis element of $W$ and places it at one position in $V$ —so the coefficient $A_{pq}$ tells you how much the $q$ -th basis vector of $V$ contributes to the $p$ -th basis vector of $W$ under $T$ . This is precisely what the matrix of $T$ records: evaluate both sides of the expansion on $\alpha_q$ and read off the coefficients of the image in basis $\mathfrak{B}'$ .

Intuitive explanation

Related theory

Change Of Basis And The Change Of Basis Matrix

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