## Section5.4Eigenvectors and Eigenspaces (GT4)

### Subsection5.4.1Class Activities

#### Activity5.4.1.

It's possible to show that $$-2$$ is an eigenvalue for $$\left[\begin{array}{ccc}-1&4&-2\\2&-7&9\\3&0&4\end{array}\right]\text{.}$$

Compute the kernel of the transformation with standard matrix

\begin{equation*} A-(-2)I = \left[\begin{array}{ccc} \unknown & 4&-2 \\ 2 & \unknown & 9\\3&0&\unknown \end{array}\right] \end{equation*}

to find all the eigenvectors $$\vec x$$ such that $$A\vec x=-2\vec x\text{.}$$

#### Definition5.4.2.

Since the kernel of a linear map is a subspace of $$\IR^n\text{,}$$ and the kernel obtained from $$A-\lambda I$$ contains all the eigenvectors associated with $$\lambda\text{,}$$ we call this kernel the eigenspace of $$A$$ associated with $$\lambda\text{.}$$

#### Activity5.4.3.

Find a basis for the eigenspace for the matrix $$\left[\begin{array}{ccc} 0 & 0 & 3 \\ 1 & 0 & -1 \\ 0 & 1 & 3 \end{array}\right]$$ associated with the eigenvalue $$3\text{.}$$

#### Activity5.4.4.

Find a basis for the eigenspace for the matrix $$\left[\begin{array}{cccc} 5 & -2 & 0 & 4 \\ 6 & -2 & 1 & 5 \\ -2 & 1 & 2 & -3 \\ 4 & 5 & -3 & 6 \end{array}\right]$$ associated with the eigenvalue $$1\text{.}$$

#### Activity5.4.5.

Find a basis for the eigenspace for the matrix $$\left[\begin{array}{cccc} 4 & 3 & 0 & 0 \\ 3 & 3 & 0 & 0 \\ 0 & 0 & 2 & 5 \\ 0 & 0 & 0 & 2 \end{array}\right]$$ associated with the eigenvalue $$2\text{.}$$

### Subsection5.4.3Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/GT4.slides.html.

### Subsection5.4.5Mathematical Writing Explorations

#### Exploration5.4.6.

Given a matrix $$A\text{,}$$ let $$\{\vec{v_1},\vec{v_2},\ldots,\vec{v_n}\}$$ be the eigenvectors with associated distinct eigenvalues $$\{\lambda_1,\lambda_2,\ldots, \lambda_n\}\text{.}$$ Prove the set of eigenvectors is linearly independent.

### Subsection5.4.6Sample Problem and Solution

Sample problem Example B.1.24.