Question

Let \(A = \left[ {\begin{array}{*{20}{c}}2&0\\0&2\end{array}} \right]\), and define \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) by \(T\left( {\bf{x}} \right) = A{\bf{x}}\). Find the images under \(T\) of \({\bf{u}} = \left[ {\begin{array}{*{20}{c}}1\\{ - 3}\end{array}} \right]\), and \({\bf{v}} = \left[ {\begin{array}{*{20}{c}}a\\b\end{array}} \right]\).

Short answer

The images under T of vectors u and v are \(T\left( {\bf{u}} \right) = \left[ {\begin{array}{*{20}{c}}2\\{ - 6}\end{array}} \right]\) and \(T\left( {\bf{v}} \right) = \left[ {\begin{array}{*{20}{c}}{2a}\\{2b}\end{array}} \right]\).

Step-by-step solution

Write the concept for computing images under the transformation of vectors

The multiplication of matrix\(A\)of the order\(m \times n\)and vector x gives a new vector defined as\(A{\bf{x}}\)or b.

This concept is defined by the transformation rule \(T\left( {\bf{x}} \right)\). The matrix transformation is denoted as \({\bf{x}}| \to A{\bf{x}}\).

Obtain the image of vector u under transformation T

Consider the transformation\(T\left( {\bf{x}} \right) = A{\bf{x}}\).

Substitute ufor x in the transformation \(T\left( {\bf{x}} \right) = A{\bf{x}}\)to obtain the image of vector u under transformation T.

\(T\left( {\bf{u}} \right) = A{\bf{u}}\)

Substitute matrix\(A = \left[ {\begin{array}{*{20}{c}}2&0\\0&2\end{array}} \right]\)and\({\bf{u}} = \left[ {\begin{array}{*{20}{c}}1\\{ - 3}\end{array}} \right]\)in\(T\left( {\bf{u}} \right) = A{\bf{u}}\), as shown below:

\(\begin{aligned}{c}T\left( {\bf{u}} \right) &= \left[ {\begin{array}{*{20}{c}}2&0\\0&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1\\{ - 3}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{2\left( 1 \right) + 0\left( { - 3} \right)}\\{0\left( 1 \right) + 2\left( { - 3} \right)}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}2\\{ - 6}\end{array}} \right]\end{aligned}\)

Thus, \(T\left( {\bf{u}} \right) = \left[ {\begin{array}{*{20}{c}}2\\{ - 6}\end{array}} \right]\).

Obtain the image of vector v under transformation T

Substitute vfor x in the transformation \(T\left( {\bf{x}} \right) = A{\bf{x}}\)to obtain the image of vector v under transformation T.

\(T\left( {\bf{v}} \right) = A{\bf{v}}\)

Substitute matrix\(A = \left[ {\begin{array}{*{20}{c}}2&0\\0&2\end{array}} \right]\)and\({\bf{v}} = \left[ {\begin{array}{*{20}{c}}a\\b\end{array}} \right]\)in\(T\left( {\bf{v}} \right) = A{\bf{v}}\), as shown below:

\(\begin{aligned}{c}T\left( {\bf{v}} \right) &= \left[ {\begin{array}{*{20}{c}}2&0\\0&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}a\\b\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{2\left( a \right) + 0\left( b \right)}\\{0\left( a \right) + 2\left( b \right)}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{2a}\\{2b}\end{array}} \right]\end{aligned}\)

Thus, \(T\left( {\bf{v}} \right) = \left[ {\begin{array}{*{20}{c}}{2a}\\{2b}\end{array}} \right]\).