Question

The figure shows vectors \(u\), \(v\) and \(w\) along with the images \(T\left( u \right)\) and \(T\left( v \right)\) under the action of a linear transformation \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\). Copy this figure carefully, and draw the image \(T\left( w \right)\) as accurately as possible. [Hint: First write \(w\) as a linear combination of \(u\) and \(v\).]

Short answer

In the first figure, draw lines through the head of \(w\), where one is parallel to \(v\) and the other parallel to \(u\).

From the figure, it can be estimated that

\(w = u + 2v\)

Step-by-step solution

Use the graph find the relation between \(u\), \(v\), and \(w\)

In the first figure, draw lines through the head of \(w\), where one is parallel to \(v\) and the other parallel to \(u\).

From the figure, it can be estimated that

\(w = u + 2v\)

Find the linear transformation

As \(T\) is linear, the transformation is:

\(\begin{aligned}T\left( w \right) &= T\left( u \right) + T\left( {2v} \right)\\ &= T\left( u \right) + 2T\left( v \right)\end{aligned}\)

Drawing the image for \(T\left( w \right)\)

Draw \(2T\left( v \right)\) in the right figure and a line parallel to \(T\left( u \right)\) through the head of \(2T\left( v \right)\) and another line parallel to \(2T\left( v \right)\) through the head of \(T\left( u \right)\).

So, the diagonal of the parallelogram in the above figure is \(T\left( w \right)\).