MATH 2660

Midterm 3 | Spring '11

1. Find the angle formed by two vectors $x = \left( \begin{array}{c} 1 \\ 2 \\ 2 \end{array} \right)$ and $y = \left( \begin{array}{c} 2 \\ 2 \\ 0 \end{array} \right)$ in $\mathbb{R}^3$.

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2. Find the least-square solution of the linear system $Ax = b$, where $A = \left( \begin{array}{c c c} 1 & 1 & 3 \\ -1 & 3 & 1 \\ 1 & 2 & 4 \end{array} \right) \quad \text{and} \quad b = \left( \begin{array}{c} -2 \\ 0 \\ 8 \end{array} \right)$

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3. Let $x_1, x_2, x_3$ be the columns of the matrix $\left[ \begin{array}{c c c} 2 & -2 & 1 \\ 1 & 5 & 11 \\ 2 & 4 & -2 \end{array} \right]$. If they form a basis of $\mathbb{R}^3$, use the Gram-Schmidt process to find an orthonormal basis $\{ u_1, u_2, u_3 \}$ of $\mathbb{R}^3$ such that $\text{Span}(u_1, \dots, u_k) = \text{Span}(x_1, \dots, x_k) \quad k = 1, 2, 3$

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4. Find the eigenvalues and corresponding eigenspaces for the matrix $A = \left( \begin{array}{c c c} 1 & 2 & 1 \\ 0 & 3 & 1 \\ 0 & 5 & -1 \end{array} \right)$

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5. Determine whether the matrix $A = \left( \begin{array}{c c c} 1 & 2 & 1 \\ 0 & 3 & 1 \\ 0 & 5 & -1 \end{array} \right)$ is diagonalizable. Explain your answer. If it is, find a matrix $X$ such that $X^{-1}AX = D$, where $D$ is diagonal.

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6. Find $A^5$ for $A = \left( \begin{array}{c c c} 4 & -5 & 1 \\ 1 & 0 & -1 \\ 0 & 1 & -1 \end{array} \right)$.

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