Linear Transformations and Diagonalization

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Section 5.3 Linear Transformations and Diagonalization

Goals.

We will

review the process of finding the matrix of a transformation,
review the process of change of basis,
diagonalize the matrix of a transformation, and
note a useful interpretation this form of a transformation.

In this section we find a connection between diagonalization and some linear transformations.

Subsection 5.3.1 Return to Matrix of a Transformation

Activity 5.3.1.

In this activity use the linear transformation $T(ax^2+bx+c)=(8a)x^2+(2b)x+(2c)\text{,}$ the standard basis $\B=\{1,x,x^2\}\text{,}$ and the additional basis $\C=\{1+x+x^2,-1+x^2,-1+x\}\text{.}$

(a)

Find the matrix of the transformation for $T$ with respect to the standard basis $\B\text{.}$

Remember this matrix maps coordinates of inputs to coordinates of outputs both with respect to $\B\text{.}$

(b)

Find the change of basis matrix $\underset{\B \leftarrow \C}{P}\text{.}$

Remember this matrix maps coordinates with respect to $\C$ to coordinates with respect to $\B\text{.}$

(c)

Find the change of basis matrix $\underset{\C \leftarrow \B}{Q}\text{.}$

(d)

Use the above matrices to find a matrix of the transformation for $T$ with respect to $\C\text{.}$

Theorem 5.3.1.

If a matrix $A$ is diagonalizable as $A=PDP^{-1}$ then $D$ is the matrix of the transformation $A\vec{x}$ with respect to the basis from the columns of $P\text{.}$

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