MTH1004 Linear Algebra Practical Class 6

Question 1

Question

Show that $S=\{\left(\begin{array}{c}x\\ x-y\\ x+2y\end{array}\right):x,y\in \mathbb{R}\}$ is the vector subspace of ${\mathbb{R}}^3$ and find a basis for $S$.

Solution

…

Question 2

Question

Which of the following sets are vector subspaces of ${\mathbb{R}}^3$ and why. (a) $S_1=\{x\in {\mathbb{R}}^3:x_1=x_2=x_3\}$ (b) $S_2=\{x\in {\mathbb{R}}^3:x_1+x_2+x_3=0\}$ (c) $S_3=\{x\in {\mathbb{R}}^3:x_1\ge 0\}$

Solution

…

Question 3

Question

Find a basis for the set of solutions of the following linear system: $n{x}_1+x_2+3x_3-x_4=0$, $x_2+4x_3+x_4=0$

Solution

…

Question 4

Question

Check whether the following vectors form a basis for ${\mathbb{R}}^3$: $u=\left(\begin{array}{c}1\\ 1\\ 2\end{array}\right)$, $v=\left(\begin{array}{c}2\\ 1\\ -1\end{array}\right)$, $w=\left(\begin{array}{c}1\\ 1\\ -2\end{array}\right)$

Solution

…

Question 5

Question

Find $\text{col}(A)$, $\text{row}(A)$, and $\text{null}(A)$ for the matrix: $A=\left(\begin{array}{ccc}1& 1& -1\\ -1& 1& 1\\ 1& -1& 1\end{array}\right)$. Find a basis for each for the above vector subspaces and determine their dimension. Verify the Rank Theorem: $\text{rank}(A)+\text{nullity}(A)=\text{dim}({\mathbb{R}}^3)$.

Solution

…