How to find a transformation matrix in a different basis

A transformation can have different matrices when we describe the inputs and outputs in different bases. The underlying map stays the same, but the coordinates change. Let’s follow those coordinate changes to build the matrix in a new basis.

Learn Linear Algebra with meStep-by-step videos, lesson notes, and practice for matrices, vectors, and transformations.

Explore Linear Algebra →

Use a basis matrix to convert coordinates

Let $B$ be the ordered basis $(1,1)$, $(-3,-2)$. Put those vectors into the columns of $C$. Multiplying $C[x{]}_B$ converts basis coordinates into standard coordinates.

$C=\left[\begin{array}{rr}1& -3\\ 1& -2\end{array}\right]$

Use the following matrix for the transformation in standard coordinates.

$A=\left[\begin{array}{rr}2& -1\\ -3& 0\end{array}\right]$

Convert to standard coordinates, apply A, then convert back

$\left[T\right(x){]}_B=C^{-1}AC[x{]}_B$

The matrix of the transformation in basis $B$ is $C^{-1}AC$.

$M=\left[\begin{array}{rr}-11& 35\\ -4& 13\end{array}\right]$

Apply the new matrix to basis coordinates

For $[x{]}_B=(2,1)$, multiply by $M$.

$\begin{array}{l}-11\left(2\right)+35\left(1\right)=13\\ -4\left(2\right)+13\left(1\right)=5\end{array}$

The output coordinates are $(13,5)$ in basis $B$. To check them, the input in standard coordinates is $C(2,1)=(-1,0)$. Applying $A$ gives $(-2,3)$, and converting $(13,5)$ back with $C$ gives the same $(-2,3)$.

Full Linear Algebra course

Learn Linear Algebra with me

We’ll work through matrices, vectors, and transformations, one step at a time.

Explore my Linear Algebra course →

• Video lessons
• Lesson notes
• Practice problems

Included with membership, along with every other Krista King Math course.