12  Gaussian elimination

Exercises

Exercise 12.1 (Gaussian elimination and overlap matrix) Consider the two vectors

\[ \mathbf{v}_1= \begin{pmatrix} 1\\ 1\\ 0 \end{pmatrix}, \qquad \mathbf{v}_2= \begin{pmatrix} 1\\ 0\\ 1 \end{pmatrix}. \]

  1. Compute the overlap matrix \(S\) with elements

    \[ S_{ij}=\mathbf{v}_i^T\mathbf{v}_j. \]

  2. Verify that \(S\) is symmetric.

  3. Use Gaussian elimination on the augmented matrix \([S\,|\,I]\) to compute \(S^{-1}\).

  4. Verify your result by computing \(SS^{-1}\).

Exercise 12.2 Write down the dual basis of \(B = \{\mathbf{b}_1,\mathbf{b}_2\}\).

Exercise 12.3 (Robert Beezer) Consider the following system of equations:

\[ \begin{aligned} 2 x_1 -3x_2 + x_3 + 7 x_4 &= 14\\ 2x_1 + 8x_2 - 4x_3 + 5x_4 &= -1\\ x_1 + 3x_2 - 3x_3 &= 4\\ -5 x_1 + 2x_2 + 3 x_3 + 4x_4 &= -19 \end{aligned} \]

Use Gaussian elimination to find all possible solutions. Write the solution set using set notation.

Exercise 12.4 (Robert Beezer) Find all possible solutions of the linear system \[ \begin{aligned} 3 x_1 + 4 x_2 - x_3 + 2 x_4 &= 6 \\ x_1 - 2 x_2 + 3 x_3 + x_4 &= 2 \\ 10 x_2 - 10 x_3 - x_4 &= 1 \end{aligned} \]

Write down the solution set using set notation.

Exercise 12.5 (Robert Beezer) Find all possible solutions of the linear system \[ \begin{aligned} 2 x_1 + 4 x_2 + 5 x_3 + 7 x_4 &= -26 \\ x_1 + 2x_2 + x_3 - x_4 &= -4 \\ -2 x_1 - 4 x_2 + x_3 + 11 x_4 &= -10 \end{aligned} \]

Write down the solution set using set notation.

Exercise 12.6 Let \(\mathbf{b}_i\), \(i=1,2,\cdots,n\) be a basis for \(\mathbb{R}^n\). Let \[B = [\mathbf{b}_1, \mathbf{b}_2, \cdots, \mathbf{b}_n]\] be the basis matrix, whose columns are precisely the \(\mathbf{b}_i\). Explain that if \(\mathbf{y} = x_1 \mathbf{b}_1 + \cdots + x_n \mathbf{b}_n\), then \[B \mathbf{x} = \mathbf{y}.\] Solve for \(\mathbf{x}\) using \(B\) in symbols.

Exercise 12.7 Let \(n\) linearly independent vectors \(B = \{\mathbf{b}_1, \cdots,\mathbf{b}_n\}\) of \(\mathbb{F}^n\) be given. Write down a formula for the dual basis in terms of the overlap matrix.