Linear Algebra Practical Class, Week 20

Question One

Question

Let $\{{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\}$ be the standard vectors in ${\mathbb{R}}^3$. Simplify the sets:

1. $\text{span}({\mathbf{e}}_1,{\mathbf{e}}_1+{\mathbf{e}}_2,{\mathbf{e}}_1+{\mathbf{e}}_2+{\mathbf{e}}_3)$.
2. $\text{span}({\mathbf{e}}_1,-2{\mathbf{e}}_2,{\mathbf{e}}_1+{\mathbf{e}}_2-{\mathbf{e}}_3)$.
3. $\text{span}({\mathbf{e}}_1+{\mathbf{e}}_2,{\mathbf{e}}_2,{\mathbf{e}}_2-3{\mathbf{e}}_1)$.

Solution

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Question Two

Question

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

Solution

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Question Three

Question

Find the span of the following vectors:

1. $\mathbf{u}=\left[\begin{array}{c}2\\ 0\\ 3\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]$.
2. $\mathbf{u}=\left[\begin{array}{c}0\\ -2\\ 0\\ 5\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}3\\ 1\\ 1\\ 0\end{array}\right]$.
3. $\mathbf{u}=\left[\begin{array}{c}1\\ -1\\ 0\\ 2\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}0\\ -\frac{1}{2}\\ \frac{3}{2}\\ 1\end{array}\right]$ and $\mathbf{w}=\left[\begin{array}{c}-1\\ 0\\ 1\\ 0\end{array}\right]$.

Solution

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Question Four

Question

Find a value for $c$ so that the following vectors are linearly independent:

1. $\mathbf{u}=\left[\begin{array}{c}1\\ 1-c\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}1\\ 0\end{array}\right]$.
2. $\mathbf{u}=\left[\begin{array}{c}1\\ c\\ 3\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}\frac{1}{3}\\ -\frac{1}{3}\\ 1\end{array}\right]$.

Solution

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Question Five

Question

Check whether the following vectors in ${\mathbb{R}}^3$ are linearly independent, and then find the span of the three vectors:

$$\mathbf{u}=\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right],\mathbf{v}=\left[\begin{array}{c}-1\\ 1\\ 0\end{array}\right],\mathbf{w}=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$$

Solution

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Question Six

Question

Find the dot product, the length, and the angle between the vectors:

1. $\mathbf{u}=\left[\begin{array}{c}1\\ -1\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}0\\ 3\end{array}\right]$.
2. $\mathbf{u}=\left[\begin{array}{c}0\\ \frac{1}{2}\\ -\frac{1}{2}\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}\frac{1}{3}\\ \frac{2}{3}\\ \frac{2}{3}\end{array}\right]$.
3. $\mathbf{u}=\left[\begin{array}{c}0\\ -\frac{1}{2}\\ 2\\ 2\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}1\\ -1\\ 1\\ -1\end{array}\right]$.

Then, verify the Cauchy-Schwarz inequality and the Triangle inequality.

Solution

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