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Ne4ueva [31]
2 years ago
9

Help with this linear algebra question

Mathematics
1 answer:
JulijaS [17]2 years ago
8 0

Check what the transformation does for each vector in the standard basis of R²<em> </em>:

<em>T</em> (1, 0) = (5, 2)

<em>T</em> (0, 1) = (-4, 0)

Now compute the weights <em>a</em>, <em>b</em> and <em>c</em>, <em>d</em> such that

<em>a</em> <em>T</em> (1, 0) + <em>b</em> <em>T</em> (0, 1) = (-2, 1)

<em>c</em> <em>T</em> (1, 0) + <em>d</em> <em>T</em> (0, 1) = (-1, 1)

\left[\begin{array}{cc|c}5&-4&-2\\2&0&1\end{array}\right]\sim\left[\begin{array}{cc|c}1&0&\frac12\\\\0&1&\frac98\end{array}\right]\implies a=\dfrac12,b=\dfrac98

\left[\begin{array}{cc|c}5&-4&-1\\2&0&1\end{array}\right]\sim\left[\begin{array}{cc|c}1&0&\frac12\\\\0&1&\frac78\end{array}\right]\implies c=\dfrac12,d=\dfrac78

Then the matrix <em>A'</em> is

A'=\begin{bmatrix}\frac12&\frac12\\\\\frac98&\frac78\end{bmatrix}

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