Preview Activity 4.3.1
Let's recall how a vector in \(\real^2\) can be represented in a coordinate system defined by a basis \(\bcal=\{\vvec_1, \vvec_2\}\text{.}\)

Suppose that we consider the basis \(\bcal\) defined by
\begin{equation*} \vvec_1 = \twovec{1}{1},\qquad \vvec_2 = \twovec{1}{0}\text{.} \end{equation*}Find the vector \(\xvec\) whose representation in the coordinate system defined by \(\bcal\) is \(\coords{\xvec}{\bcal} = \twovec{3}{2}\text{.}\)
Consider the vector \(\xvec=\twovec{4}{5}\) and find its representation \(\coords{\xvec}{\bcal}\) in the coordinate system defined by \(\bcal\text{.}\)
How do we use the matrix \(C_{\bcal} = \left[\begin{array}{rr} \vvec_1 \amp \vvec_2 \end{array}\right]\) to convert a vector's representation \(\coords{\xvec}{\bcal}\) in the coordinate system defined by \(\bcal\) into its standard representation \(\xvec\text{?}\) How do we use this matrix to convert \(\xvec\) into \(\coords{\xvec}{\bcal}\text{?}\)
Suppose that we have a matrix \(A\) whose eigenvectors are \(\vvec_1\) and \(\vvec_2\) and associated eigenvalues are \(\lambda_1=4\) and \(\lambda_2 = 2\text{.}\) Express the vector \(A(3\vvec_1 +5\vvec_2)\) as a linear combination of \(\vvec_1\) and \(\vvec_2\text{.}\)
If \(\coords{\xvec}{\bcal} = \twovec{3}{5}\text{,}\) find \(\coords{A\xvec}{\bcal}\text{.}\)