Let K = \begin{pmatrix} 5 & -3 \\ 2 & -2 \end{pmatrix}( 5 2 −3 −2 ) be considered as a linear transformation R^2R 2 ? R^2R 2 . Find the matrix L representing the same transformation with respect to the basis { b1 = \begin{pmatrix} 3 \\ 1 \end{pmatrix}( 3 1 ) , b2 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}( 1 2 ) } .
Answers
Answer:
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Answer:
To find the matrix representation of the linear transformation K with respect to the given basis, we need to find the images of the basis vectors under the transformation and then express these images as linear combinations of the basis vectors.
Explanation:
First, we apply the transformation K to the basis vectors b1 and b2:
K(b1) = \begin{pmatrix} 5 & -3 \ 2 & -2 \end{pmatrix} \begin{pmatrix} 3 \ 1 \end{pmatrix} = \begin{pmatrix} 12 \ 4 \end{pmatrix} = 4 \begin{pmatrix} 3 \ 1 \end{pmatrix} + (-2) \begin{pmatrix} 1 \ 2 \end{pmatrix} = 4b1 - 2b2
K(b2) = \begin{pmatrix} 5 & -3 \ 2 & -2 \end{pmatrix} \begin{pmatrix} 1 \ 2 \end{pmatrix} = \begin{pmatrix} -1 \ 0 \end{pmatrix} = (-1) \begin{pmatrix} 3 \ 1 \end{pmatrix} + 1 \begin{pmatrix} 1 \ 2 \end{pmatrix} = -b1 + b2
Next, we represent these images as column vectors with respect to the given basis:
[K(b1)]_B = \begin{pmatrix} 4 \ -2 \end{pmatrix}
[K(b2)]_B = \begin{pmatrix} -1 \ 1 \end{pmatrix}
Finally, we construct the matrix L with these column vectors as its columns:
L = \begin{pmatrix} [K(b1)]_B & [K(b2)]_B \end{pmatrix} = \begin{pmatrix} 4 & -1 \ -2 & 1 \end{pmatrix}
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