[Math] Find the Matrix A of the Linear Transformation

linear algebramatricestransformation

Can anyone walk me through the steps to complete this problem? I am unsure of where to start to solve the problem. I get that the resulting matrix $A$ should be a $2 \times2$ matrix, should I be finding another vector in the basis to find $BAB^{-1}$?

Let $V$ be the plane with equation $x_1 – 4 x_2 + 2 x_3 =0$ in ${\mathbb R}^3$. Find the matrix $A$ of the linear transformation $T(x)= \displaystyle\left[\begin{array}{ccc} -6 &-12 &0 \cr -2 &-2 &1 \cr -1 &2 &2 \cr \end{array}\right] x$ with respect to the basis $\displaystyle\left[\begin{array}{c} 4 \cr 1 \cr 0 \cr \end{array}\right]$ , $\displaystyle\left[\begin{array}{c} -2 \cr 0 \cr 1 \cr \end{array}\right]$

Best Answer

Let $u=(4,1,0)^T$ and $v=(-2,0,1)^T$.

  1. Before answering the question, do a sanity check first. Do $\mathbf{u}$ and $\mathbf{v}$ really lie on $V$? That is, do they satisfy the equation $x_1-4x_2+2x_3=0$? They seem so. Good. Let's proceed.
  2. Now compute $\mathbf{p} = T(\mathbf{u})$.
  3. Can you express $\mathbf{p}$ as a linear combination of $\mathbf{u}$ and $\mathbf{v}$? That is, can you find two numbers $a$ and $b$ such that $\mathbf{p}=a\mathbf{u}+b\mathbf{v}$? Note that no difficult calculation is needed; $a$ and $b$ can be found by simple inspection (hint: observe that $\mathbf{u}$ and $\mathbf{v}$ each has a zero entry). Verify that the $a$ and $b$ you found are correct.
  4. Similarly, find $c$ and $d$ such that $T(\mathbf{v})=c\mathbf{u}+d\mathbf{v}$.
  5. Now you may write out the matrix $A$ in terms of $a,b,c$ and $d$.
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