Linear Algebra Tutorial Class, Week 21

Question One

Question

Complete the following questions using Gauss-Jordan elimination:

1. Determine whether $\mathbf{u}=\left[\begin{array}{c}9\\ 0\\ 4\end{array}\right]$ belongs to $\text{span}(\mathbf{v},\mathbf{w},\mathbf{z}):$ $\mathbf{v}=\left[\begin{array}{c}1\\ 2\\ 4\end{array}\right],\mathbf{w}=\left[\begin{array}{c}2\\ -1\\ -1\end{array}\right],\mathbf{z}=\left[\begin{array}{c}-3\\ 1\\ 1\end{array}\right]$

2. Show that the following vectors are linearly dependent, and then simplify their span in terms of $\mathbf{u},\mathbf{v},\mathbf{w}:$ $\mathbf{u}=\left[\begin{array}{c}1\\ -1\\ 2\end{array}\right],\mathbf{v}=\left[\begin{array}{c}-3\\ 2\\ 4\end{array}\right],\mathbf{w}=\left[\begin{array}{c}-2\\ 1\\ 6\end{array}\right]$

Solution

…

Question Two

Question

$$(a)\{\begin{array}{l}kx+2y=3\\ 2x-4y=-6\end{array},(b)\{\begin{array}{l}x+ky=1\\ kx+y=1\end{array}$$

Find a value of $k$ to satisfy the following constraints for the above systems:

1. Infinitely many solutions: write the set of solutions.
2. No solutions.
3. A unique solution.

Solution

…