If L:R4 → R' is given by L(C1, C2, C3, 24) = (21 +22, 23 +...

Question

Question

If L:R4 → R is given by L(C1, C2, C3, 24) = (21 +22, 23 + 24,0), then the matrix of L with respect to the standard bases of

math algebra

Answer 1

Answer #1

Given L ( x1, x2, x3, x4 ) = ( x1 + x2, x3 + x4, 0 )

The standard basis for R4 is { (1,0,0,0) , (0,1,0,0) , (0,0,1,0) , (0,0,0,1) }

The standard basis for R3 is { (1,0,0) , (0,1,0) , (0,0,1) }

L( 1,0,0,0 ) = ( 1,0,0 ) = 1(1,0,0) + 0(0,1,0) + 0(0,0,1)

L(0,1,0,0) = (1,0,0) = 1(1,0,0) + 0(0,1,0) + 0(0,0,1)

L(0,0,1,0) = (0,1,0) = 0(1,0,0) + 1(0,1,0) +0(0,0,1)

L(0,0,0,1) = (0,1,0) = 0(1,0,0) + 1(0,1,0) + 0(0,0,1)

Writing all the coefficients in column yields

$\begin{bmatrix} 1 & 1& 0 &0 \\ 0& 0 &1 &1\\ 0&0 & 0& 0 \end{bmatrix}$

Answer 2

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