MTH1004 Linear Algebra Practical Class 3

Question 1

Question

Find the solution set which corresponds to the following row echelon forms: (i) $\left[\begin{array}{cccccc}1& -2& -1& 0& 2& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right]$ (ii) $\left[\begin{array}{cccccc}1& 0& 1& 2& 0& 3\\ 0& 0& 0& 0& 1& -1\end{array}\right]$

Solution

…

Question 2

Question

Solve the system with 3 equations and 4 unknowns using Gauss-Jordan elimination: $\{\begin{array}{l}{x}_1+2x_2-2x_3+2x_4=1\\ 2x_1-4x_2+4x_3+6x_4=0\\ x_1+2x_2-2x_3+3x_4=0\end{array}$ and write its set of solutions.

Solution

…

Question 3

Question

Let $v_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],v_2=\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right],v_3=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$ vectors in ${\mathbb{R}}^3$. Using Gauss-Jordan elimination: (i) Show that the vectors are linearly dependent. (ii) Determine whether the vector $u=\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]$ belongs to the span of the vectors $v_1,v_2,v_3$.

Solution

…

Question 4

Question

Consider the following matrices $B_1=\left[\begin{array}{ccc}-1& 0& 1\\ 0& 2& 1\\ 0& 0& 5\\ 0& 0& -1\end{array}\right],B_2=\left[\begin{array}{cccc}1& 2& 0& 5\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right],B_3=\left[\begin{array}{cccc}1& -1& 0& 1\\ 0& 0& 0& 0\end{array}\right],B_4=\left[\begin{array}{ccccc}1& 0& 0& -2& 5\\ 0& 1& 0& 9& -2\\ 0& 0& 1& 4& 3\end{array}\right]$ Find the matrices which are in row-echelon form. Which of those matrices are in reduced row-echelon form?

Solution

…