MTH1004 Linear Algebra Practical Class 2

Question 1

Question

Let $\{e_1,e_2,e_3\}$ be the standard vectors in ${\mathbb{R}}^3$. Simplify the sets: (i) $\text{span}(e_1,e_1+e_2,e_1+e_2+e_3)$ (ii) $\text{span}(e_1,-2e_2,e_1+e_2-e_3)$ (iii) $\text{span}(e_1+e_2,e_2,e_2-3e_1)$

Solution

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Question 2

Question

If $u,v,w$ are linearly independent vectors, determine whether $u$ can be expressed as a linear combination of $v,w$.

Solution

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Question 3

Question

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

Solution

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Question 4

Question

Find a value for $c$ so that the following vectors: (i) $u=\left(\begin{array}{c}1\\ 1\\ -c\end{array}\right)$, $v=\left(\begin{array}{c}1\\ 0\end{array}\right)$ (ii) $u=\left[\begin{array}{ccc}1& c& 3\end{array}\right]$, $v=\left[\begin{array}{ccc}1/3& -1/3& 1\end{array}\right]$, are linearly independent.

Solution

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Question 5

Question

Check whether the following vectors in ${\mathbb{R}}^3$ are linearly independent: $u=\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]$, $v=\left[\begin{array}{c}-1\\ 1\\ 0\end{array}\right]$, $w=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$, and find $\text{span}(u,v,w)$.

Solution

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Question 6

Question

Find the dot product, the length, and the angle between the vectors: (i) $u=\left(\begin{array}{c}1\\ -1\end{array}\right)$, $v=\left(\begin{array}{c}0\\ 3\end{array}\right)$, (ii) $u=\left[\begin{array}{ccc}0& 1/2& -1/2\end{array}\right]$, $v=\left[\begin{array}{ccc}1/3& 2/3& 2/3\end{array}\right]$, (iii) $u=\left[\begin{array}{cccc}0& -1/2& 2& 2\end{array}\right]$, $v=\left[\begin{array}{cccc}1& -1& 1& -1\end{array}\right]$. Verify the Cauchy-Schwarz inequality and the Triangle inequality.

Solution

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