2. Consider the linear system:

\(\begin{array}{r}

x+2 y+z=a \\

x+y+a z=1 \\

3 x+4 y+\left(a^{2}-5\right) z=1

\end{array}\)

For which value(s) of \(a\) does the system have NO solution?

3. Let \(v \in \mathbb{R}^{n}\) be a nonzero vector. Consider the linear transformation

\(T(x)=(v \cdot x) v\) (a) Describe the kernel of \(T\). When (in terms of \(v\) and \(n\) ) is \(T\) one-to-one?

(b) Describe the image of \(T\). When is \(T\) onto?

4. Suppose \(\mathrm{n}\) is even. Consider \(P_{n}\), the space of polynomials of degree less than or equal to \(n\). For each of the following prove or disprove that the given set is a subspace of \(P_{n}\). If the answer is yes, find a basis of the subspace.

1. \(\left\{a_{0}+a_{1} t+\cdots+a t^{n} \mid a_{0}+a_{1}+\cdots+a_{n}<1\right\}\)
2. \(\left\{f \in P_{n} \mid f(-x)=-f(x)\right.\) for all \(\left.x\right\}\)

5. Consider the linear transformation \(T: \mathbb{R}^{2 \times 2} \rightarrow \mathbb{R}^{2 \times 2}\)

\(T(M)=\left[\begin{array}{ll}

1 & 1 \\

1 & 1

\end{array}\right] M-M\left[\begin{array}{ll}

1 & 1 \\

1 & 1

\end{array}\right]\)

Write the matrix of \(T\) with respect to the basis

\(\left[\begin{array}{cc}

1 & 1 \\

-1 & -1

\end{array}\right],\left[\begin{array}{ll}

1 & -1 \\

1 & -1

\end{array}\right],\left[\begin{array}{ll}

1 & 0 \\

0 & 1

\end{array}\right],\left[\begin{array}{ll}

0 & 1 \\

1 & 0

\end{array}\right]\)

6. Compute the singular value decomposition of

\(A=\left[\begin{array}{cc}

1 & 0 \\

0 & 2 \\

2 & -1

\end{array}\right]\)

7. Find an orthonormal basis for the space spanned by the three linearly independent vectors

\(\left[\begin{array}{l}

1 \\

0 \\

0 \\

1

\end{array}\right],\left[\begin{array}{c}

1 \\

-1 \\

1 \\

1

\end{array}\right],\left[\begin{array}{c}

1 \\

0 \\

1 \\

-1

\end{array}\right]\)

8. Find all least square solutions of the following system of equations

\(\begin{aligned}

& x+y+2z=4 \\

& 2x+2y+3z=2 \\

& 3x+3y+4z=3 \\

\end{aligned}\)

9. Decide whether the following statement is true or false: If \(A\) is a symmetric \(n \times n\) matrix and there is some positive integer \(k\) so that \(A^{k}=0\) (the zero matrix), then \(A\) must be the zero matrix. Give a full explanation of your answer.

10. Let \(A=\left[\begin{array}{cc}-15 & 28 \\ -8 & 15\end{array}\right]\)

1. Find all eigenvalues of \(A\) and compute an eigenvector for each one.
2. Find an invertible matrix \(P\) and a diagonal matrix \(D\) such that \(A=P^{-1} D P\)
3. Compute \(A^{37}\)

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