<s>[INST] Find bases for eigenspaces of A $$A = \begin{pmatrix} 6 & 4 \\ -3 & -1\end{pmatrix}$$
Find the bases for eigenspaces $E_{\lambda_1}$ and $E_{\lambda_2}$ of $A$.
I don't really know where to start on this problem.
 [/INST] Consider the matrix $$\lambda I - A = \begin{pmatrix}\lambda-6 & -4 \\ 3 & \lambda +1\end{pmatrix}$$
Now, our eiegnavlues of $A$, are the solution to the equation $\det{(\lambda I - A)}=0$
\begin{align}\det{(\lambda I - A)}&=0 \\ \implies(\lambda -6)(\lambda+1)-(3)(-4) &=0 \\ \implies \lambda^2-5\lambda +6 &= 0 \\\implies(\lambda-2)(\lambda-3) &=0 \\ \implies \lambda_1=2 \text{ and } \lambda_2=3\end{align}
Thus $\lambda_1=2$ and $\lambda_2=3$ are the two eigenvalues corresponding to matrix $A$.
Since we have two distinct eigenvalues, we know that we will have two eigenvectors of $A$, $\bar{x}= \begin{pmatrix}x_1 \\ x_2\end{pmatrix}$, that corresponds to each eigenvector.
To find these eigenvectors, we must have that is satisfies the equation $(\lambda I - A)\bar{x}=\bar{0}$
For $\lambda_1 =2$: $$(2I-A)=\begin{pmatrix}-4 & -4 \\ 3 &3\end{pmatrix}$$
Which, reduces to $$\begin{pmatrix}1 &1 \\ 0 &0\end{pmatrix}$$
Thus we know $x_1 +x_2 = 0$. Let $x_2=t, \ t\in \mathbb{R} \implies x_1= -t$
Thus $\bar{x}= \begin{pmatrix}x_1 \\ x_2\end{pmatrix} = t \begin{pmatrix}-1 \\ 1\end{pmatrix}, t \in \mathbb{R}$ is the eigenvector of $A$ corresponding to the eigenvalue $\lambda_1 =2$.
Thus the basis for the eigenspace of $A$ corresponding to $\lambda_1 = 2$, is given by $$E_{\lambda_1}=\bigg \{  \begin{pmatrix} -1 \\ 1\end{pmatrix} \bigg \}$$
For $\lambda_2 = 3$:
$$(3I - A) = \begin{pmatrix}-3 & -4 \\ 3 & 4\end{pmatrix}$$
which reduces to $$\begin{pmatrix}1 & \frac{4}{3} \\ 0 & 0\end{pmatrix}$$
Thus we know $x_1 + \frac{4}{3}x_2 =0$. Let $x_2 = s, s\in \mathbb{R} \implies x_1 = -\frac{4}{3}s$
Thus $\bar{x}= \begin{pmatrix}x_1 \\ x_2\end{pmatrix} = s\begin{pmatrix}-\frac{4}{3} \\ 1\end{pmatrix}$ is the eigenvector of $A$ corresponding to the eigenvector $\lambda_2=3$.
Thus the basis for the eigenspace of $A$ corresponding to eigenvalue $\lambda_2 =3$ is:
$$E_{\lambda_2}=\bigg\{ \begin{pmatrix}-\frac{4}{3} \\ 1\end{pmatrix}\bigg \}$$
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