MTH1004 Linear Algebra Tutorial Class 1

Question 1

Question

Let $\{e_1,e_2,e_3\}$ be the standard vectors in ${\mathbb{R}}^3$. Determine whether $\text{span}(e_1,-2e_2,e_1+e_2-e_3)={\mathbb{R}}^3$.

Solution

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Question 2

Question

Find the span of the following vectors: (i) $u=\left[\begin{array}{c}0\\ -2\\ 0\\ 5\end{array}\right]$, $v=\left[\begin{array}{c}3\\ 1\\ 1\\ 0\end{array}\right]$, (ii) $u=\left[\begin{array}{c}1\\ -1\\ 0\\ 2\end{array}\right]$, $v=\left[\begin{array}{c}0\\ -\frac{1}{2}\\ \frac{3}{2}\\ 1\end{array}\right]$, $w=\left[\begin{array}{c}-1\\ 0\\ 1\\ 0\end{array}\right]$

Solution

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Question 3

Question

Find a value for $c$ so that the following vectors: (i) $u=\left(\begin{array}{c}1\\ 1\end{array}\right)$, $v=\left(\begin{array}{c}1\\ 0\end{array}\right)$ are linearly independent.

Solution

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Question 4

Question

Find the dot product, the length and the angle between the vectors: $u=\left[\begin{array}{c}0\\ \frac{1}{2}\\ -\frac{1}{2}\end{array}\right]$, $v=\left[\begin{array}{c}\frac{1}{3}\\ \frac{2}{3}\\ \frac{2}{3}\end{array}\right]$. Verify the Cauchy-Schwarz inequality and the triangle inequality.

Solution

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