5. (10%) Let \( \mathbf{D} \) be an \( m \times n \) matrix of rank \( n \). Is the matrix \( \mathbf{D}^{\mathrm{T}}\mathbf{D} \) symmetric positive definite? Explain your answer.

1. (10%) Use Gaussian Elimination to solve the following linear system:$$\begin{array}{l} x{1} + x{2} + x{3} + x{4} + x{5} = 1 \\ -x{1} - x{2} + x{3} = -1 \\ -2x{1} - 2x{2} + 3x{3} = 1 \\ x{3} + x{4} + 3x{5} = -1 \\ x{1} + x{2} + 2x{3} + 2x{4} + 4x_{5} = 1 \end{array}$$

(a) (8%) Find all solutions to the system.

(b)(8%) If $$a{1} = \begin{bmatrix} 1 \\ 1 \\ 3 \\ 4 \end{bmatrix}$$ and $$a{2} = \begin{bmatrix} 2 \\ -1 \\ 1 \\ 3 \end{bmatrix}$$, determine b.

3. (12%) Let \( A = \begin{bmatrix} 2 & 1 & 1 \\ 6 & 4 & 5 \\ 4 & 1 & 3 \end{bmatrix} \). Find elementary matrices \( E_1, E_2, E_3 \) such that \( E_1 E_2 E_3 A = U \) where \( U \) is an upper triangular matrix.

(a) \( (7\%) \) Find the transition matrix from \( \{\mathbf{v}1,\mathbf{v}2,\mathbf{v}3\} \) to \( \{\mathbf{u}1,\mathbf{u}2,\mathbf{u}3\} \).

(b) \( (5\%) \) If \( x = 3v{1} + 2v{2} - v{3} \), determine the coordinates of \( x \) with respect to \( \{u{1}, u{2}, u{3}\} \).

5. (20%) Let D be the differentiation operator on the set of all polynomials of degree less than 3, and D is defined as follows:$$D(p(x)) = xp'(x) + p''(x).$$Now define the matrix A representing D with respect to $$[1, x, x^2]$$ and the matrix B representing D with respect to $$[1, x, 1+x^2]$$. Show that the matrix A and the matrix B are similar to each other.

6. (15%) Find the least squares solution of the following system.$$\begin{cases} -\frac{1}{5}x1 + \frac{1}{5}x2 = 2 \\ \frac{2}{5}x1 + \frac{1}{5}x2 = 1 \\ \frac{1}{5}x1 - \frac{2}{5}x2 = 4 \end{cases}$$

7. (15%) Give the following matrix:$$\mathbf{A} = \begin{bmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{bmatrix}.$$Show that A is positive definite based on the LU factorization of A.

1. (10%) Let $$u1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, u2 = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, u3 = \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}, b1 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}, b2 = \begin{bmatrix} 2 \\ -1 \end{bmatrix}$$. Let L be the linear transformation defined by $$L(x1, x2, x3) =$$$$(x1+x2, x1-x3)$$. Find the matrix A representing L with respect to the bases $$\{u1, u2, u3\}$$ and $$\{b1, b_2\}$$.

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