Question

Let \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) be the linear transformation such that \(T\left( {{e_1}} \right)\) and \(T\left( {{e_2}} \right)\) are the vectors shown in the figure. Using the figure, sketch the vector \(T\left( {2,1} \right)\).

Short answer

Using linear transformation \(\left( {2,1} \right)\) can be written as follows:

\(\left( {2,1} \right) = 2{e_1} + {e_2}\)

Step-by-step solution

Write \(T\left( {2,1} \right)\) in the form of \({e_1}\) and \({e_2}\)

Using linear transformation \(\left( {2,1} \right)\) can be written as follows:

\(\left( {2,1} \right) = 2{e_1} + {e_2}\)

Using linearity of \(T\) find image of \(\left( {2,1} \right)\)

\(\begin{aligned} T\left( {2,1} \right) &= T\left( {2{e_1}} \right) + T\left( {{e_2}} \right)\\ &= 2T\left( {{e_1}} \right) + T\left( {{e_2}} \right)\end{aligned}\)

Locate \(2T\left( {{e_1}} \right)\) and \(T\left( {{e_2}} \right)\) in the graph

In the given graph, locate \(2T\left( {{e_1}} \right)\) and \(T\left( {{e_2}} \right)\) to form a parallelogram as shown below: