16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.5.9

Linear Transformations

Mathematics — Linear Algebra · annihilator-matrix-subspace

Problem

Let

V

be the vector space of all

2 \times 2

matrices over the field of real numbers, and let

B = \begin{bmatrix} 2 & -2 \\ -1 & 1 \end{bmatrix}.

Let

W

be the subspace of

V

consisting of all

A

such that

AB = 0

. Let

f

be a linear functional on

V

which is in the annihilator of

W

. Suppose that

f(I) = 0

and

f(C) = 3

, where

I

is the

2 \times 2

identity matrix and

C = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}.

Find

f(B)