16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.4.13

Linear Transformations

Mathematics — Linear Algebra · basis-for-transformations-and-matrix

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}'