Eigenvalues and Orthogonality: math test, College – download the PDF

1 Eigenvalues and eigenvectors / 4 pts

Let \(A=\begin{pmatrix}7&2\\3&8\end{pmatrix}\).

1. Find the characteristic polynomial of \(A\).
2. Find the eigenvalues of \(A\).
3. Find an eigenvector for each eigenvalue.

2 Diagonalization / 3 pts

Let \(M=\begin{pmatrix}3&0&0\\0&1&2\\0&2&1\end{pmatrix}\).

1. Find the eigenvalues of \(M\).
2. Is \(M\) diagonalizable? Justify.

3 Dot product and angle / 4 pts

Let \(\mathbf{u}=(3,-4,12)\) and \(\mathbf{w}=(2,1,5)\).

1. Compute \(\|\mathbf{u}\|\).
2. Compute \(\mathbf{u}\cdot\mathbf{w}\) and the angle between the vectors to the nearest tenth of a degree (calculator allowed).
3. Find \(t\) so that \(\mathbf{u}\) is orthogonal to \((t,-3,2)\).

4 Orthogonal projection / 3 pts

Let \(\mathbf{a}=(2,-1)\) and \(\mathbf{b}=(7,1)\).

1. Compute the projection \(\mathbf{p}\) of \(\mathbf{b}\) onto the line spanned by \(\mathbf{a}\).
2. Verify that \(\mathbf{b}-\mathbf{p}\) is orthogonal to \(\mathbf{a}\), then find the distance from \(\mathbf{b}\) to the line (calculator allowed).

5 Gram-Schmidt / 3 pts

Apply Gram-Schmidt to \(\mathbf{v}_1=(2,1,2)\) and \(\mathbf{v}_2=(3,3,0)\), then give an orthonormal pair.

6 Least squares / 3 pts

A bike rental shop records the number of bikes rented (in tens) over four days: day \(0\): \(3\); day \(1\): \(5\); day \(2\): \(6\); day \(3\): \(10\). Find the least squares line \(y=a+bx\) and use it to predict the value for day \(4\) (calculator allowed).