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Let $L : \mathbb{R}^2 \to \mathbb{R}^2 $ be a linear transformation whose matrix in the standard basis is $$\mathrm{A}= \begin{bmatrix} 2 & -1 \\ 3 & 6 \\ \end{bmatrix} $$

  1. Determine the characteristic polynomial $C_A(\lambda)$, and find the roots of $C_A(\lambda)=0$ .

    • Let $\lambda_1$ and $\lambda_2$ denote these two roots. Find an eigenvector $V_1$ belonging to $\lambda_1$. Find an eigenvector $V_2$ belonging to $\lambda_2$.
    • Show that $V_1$ and $V_2$ are linearly independent. Thus $E = \left[V_1,V_2 \right]$ is a basis for $\mathbb{R}^2$.
    • Find $\mathrm{B}$, the matrix of the linear transformation $L$ in the basis $E$. Find an invertible matrix $\mathrm{P}$ such that $\mathrm{B} = \mathrm{P}^{−1}\cdot \mathrm{A}\cdot\mathrm{P}$.
  2. Give an example of a $3 \times 3$ matrix whose characteristic polynomial is $(\lambda − 5)*3$, and such that the space of eigenvectors with eigenvalue $5$ is one-dimensional.

    • Give a second example with the same characteristic polynomial, and such that the space of eigenvectors with eigenvalue $5$ is two dimensional. Give a third example where the space of eigenvectors is three dimensional.

My attempt:

  1. I found the $C_A$ which is $ \ \lambda^2-8\lambda+15$ and the roots are $3,5$. The eigenvectors are $V_1 = (-1, 3)$ and $v_2 = (-1, 1)$.

    • I need help on how to find $V_1$ and $V_2$ are linearly independent and finding the matrix $\mathrm{B}$. The last part I have some understanding but I'm not completely sure of which is $\mathrm{B} = \mathrm{P}^{−1}\cdot \mathrm{A}\cdot\mathrm{P}$.

    • I also need help on Part $2.$ on finding examples of matrices that are $1D$ $2D$ and $3D$. Any help is appreciated. If you guys could show your work step by step I would love it! Thank you for even viewing this problem!

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Vectors are linearly independent if they are not linearly dependent. Vectors are linearly dependent if one or more of them can be written as the sum of multiples of other vectors in the set.

So, in other words, $n$ vectors are linearly independent if $v_k \neq \sum_{\substack{i=1}{i\neq k}}^n c_i v_i$ for any set of $c_i$ and for any $k$.

In even more plain language: if you can write one vector as the sum of some others, then that vector is linearly dependent on the set of the others.

So, the vector $\begin{pmatrix} 1 \\ 1 \\ 0\end{pmatrix}$ is linearly dependent on $\begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix}$ and $\begin{pmatrix} 0 \\ \frac12 \\ 0\end{pmatrix}$ (one times the first plus two times the second).

When you have only two 2-D vectors, then they're linearly independent so long as one is not a direct multiple of the other. This should be extremely easy to check. Can you find some constant $c$ such that $$\begin{pmatrix} -1 \\ 3\end{pmatrix} = c \begin{pmatrix} -1 \\ 1 \end{pmatrix}?$$

If not, then they are linearly independent.

That should get you started.

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