MATH347: Linear algebra for applications

Solve the following problems (5 course points each). Present a brief motivation of your method of solution. Problems 9 and 10 are optional; attempt them if you wish to improve your midterm examination score.

1. State the matrix product to obtain 3 linear combinations of vectors

   \(\displaystyle \boldsymbol{u}= \left[ \begin{array}{l} 1\\ 0\\ 1 \end{array} \right], \boldsymbol{v}= \left[ \begin{array}{l} - 1\\ 0\\ 1 \end{array} \right],\)

   with scaling coefficients \((\alpha_1, \beta_1) = (1, 1)\), \((\alpha_2, \beta_2) = (- 1, 1)\), \((\alpha_3, \beta_3) = (1, - 1)\).

2. Orthonormalize the vectors

   \(\displaystyle \boldsymbol{u}= \left[ \begin{array}{l} 1\\ 0\\ 1 \end{array} \right], \boldsymbol{v}= \left[ \begin{array}{l} - 1\\ 0\\ 1 \end{array} \right], \boldsymbol{w}= \left[ \begin{array}{l} 1\\ 1\\ - 1 \end{array} \right] .\)

3. For \(x, y \in \mathbb{R}\), expansion of \((x - y)^3\) leads to \((x - y)^3 = x^3 - 3 x^2 y + 3 x y^2 - y^3\). Find the corresponding expansion of \((\boldsymbol{A}-\boldsymbol{B})^3\) for \(\boldsymbol{A}, \boldsymbol{B} \in \mathbb{R}^{m \times m}\).

4. Find the projection of \(\boldsymbol{b}\) onto \(C (\boldsymbol{A})\) for

   \(\displaystyle \boldsymbol{b}= \left[ \begin{array}{l} 1\\ 2\\ 3 \end{array} \right], \boldsymbol{A}= \left[ \begin{array}{ll} 1 & - 1\\ 1 & 0\\ 1 & 1 \end{array} \right] .\)

5. Find the \(L U\) decomposition of

   \(\displaystyle \boldsymbol{A}= \left[ \begin{array}{lll} 1 & 1 & 1\\ 2 & 3 & 3\\ 3 & 5 & 6 \end{array} \right] .\)

6. State the eigenvalues and eigenvectors of \(\boldsymbol{R} \in \mathbb{R}^{2 \times 2}\), the matrix describing reflection across the vector \(\boldsymbol{w}= \left[ \begin{array}{ll} 1 & 2 \end{array} \right]^T\).

7. Compute the eigendecomposition of

   \(\displaystyle \boldsymbol{A}= \left[ \begin{array}{lll} 5 / 2 & 0 & 1 / 2\\ 0 & 1 & 0\\ 1 / 2 & 0 & 5 / 2 \end{array} \right] .\)

8. Find the SVD of

   \(\displaystyle \boldsymbol{A}= \left[ \begin{array}{ll} 1 & - 1\\ 0 & 0\\ 1 & 1 \end{array} \right] .\)

9. Find the matrix of the reflection of \(\mathbb{R}^2\) vectors across the vector \(\boldsymbol{u}= \left[ \begin{array}{ll} 1 & 2 \end{array} \right]^T\).

10. Find bases for the four fundamental spaces of

    \(\displaystyle \boldsymbol{A}= \left[ \begin{array}{ll} 1 & - 1\\ 0 & 0\\ 1 & 1 \end{array} \right] .\)