Question

Let V = W = R2. Choose the basis B = {x1, x2} of V ,...

Let V = W = R2. Choose the basis B = {x1, x2} of V , where x1 = (2, 3), x2 = (4,−5) and choose the basis D = {y1,y2} of W, where y1 = (1,1), y2 = (−3,4). Find the matrix of the identity linear mapping I : V → W with respect to these bases.

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Answer #1

Let A =

1

-3

2

4

1

4

3

-5

The RREF of A is

1

0

17/7

1/7

0

1

1/7

-9/7

Hence (2,3) = (17/7)(1,1)+(1/7)(-3,4) and (4,-5) = (1/7)(1,1)-(9/7)(-3,4).

Let I: V →W be the identity linear mapping. Then I( x1) = I(2,3) = (2,3) = (17/7)(1,1)+(1/7)(-3,4) and I ( x2) = I(4,-5) = (4,-5)= (1/7)(1,1)-(9/7)(-3,4).

Thus, the matrix of the identity linear mapping I : V →W with respect to the bases B and D is M =

17/7

1/7

1/7

-9/7
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