Linear Algebra Practical Class, Week 19

Question One

Question

Determine whether the following vectors are linearly independent:

1. $\vec{u}=\left[\begin{array}{c}3\\ 1\end{array}\right],\vec{v}=\left[\begin{array}{c}0\\ -1\end{array}\right]$.

2. $\vec{u}=\left[\begin{array}{c}2\\ -1\end{array}\right],\vec{v}=\left[\begin{array}{c}-1\\ -1\end{array}\right]$.

Solution

$$\begin{array}{rl}\left[\begin{array}{c}0\\ 0\end{array}\right]& =c_1\left[\begin{array}{c}3\\ 1\end{array}\right]+c_2\left[\begin{array}{c}0\\ -2\end{array}\right]\\ & =\left[\begin{array}{c}3c_1\\ c_1\end{array}\right]+\left[\begin{array}{c}0\\ -2c_2\end{array}\right]\end{array}$$

Thus the only solution is $c_1=0$ and $c_2=0$, therefore the two vectors are linearly independent.

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$$\begin{array}{rl}\left[\begin{array}{c}0\\ 0\end{array}\right]& =c_1\left[\begin{array}{c}2\\ -1\end{array}\right]+c_2\left[\begin{array}{c}-1\\ -1\end{array}\right]\\ & =\left[\begin{array}{c}2c_1\\ -c_1\end{array}\right]+\left[\begin{array}{c}-c_2\\ -c_2\end{array}\right]\end{array}$$

Thus $c_1=-c_2⟹0=3c_1$, therefore the only solution is $c_1=0$ and $c_2=0$ such that both vectors are linearly independent.

Question Two

Question

Find a value for $x$ so that the following vectors are linearly dependent, and another value for which they are linearly independent:

$\vec{u}=\left[\begin{array}{c}1\\ 1\end{array}\right],\vec{v}=\left[\begin{array}{c}1\\ x\end{array}\right]$

Solution

Let $c_1,c_2\in \mathbb{R}$ satisfying $c_1\mathbf{u}+c_2\mathbf{v}=0⟹$

$$\left[\begin{array}{c}{c}_1+c_2\\ c_1+x\cdot c_2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]⟹c_2(x-1)=0$$

Thus, the vectors are linearly dependent if $x=1$ as the two vectors are then multiples of one another and there are infinite solutions to the system of equations.

Alternatively, the vectors are linearly independent if $x\ne 1$, for example $x=2$, as the solution to the system of equations would be where $c_1=c_2=0$.

Question Three

Question

Find two nontrivial vectors (other than ${\vec{e}}_1,{\vec{e}}_2$) that span the plane.

Solution

The are an infinite number of vectors that span the 2D plane, other than the trivial vectors ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$, and simply include any two linearly independent vectors. For example,

$${\mathbf{v}}_1=\left[\begin{array}{c}2\\ 1\end{array}\right],{\mathbf{v}}_2=\left[\begin{array}{c}-1\\ 2\end{array}\right]$$

Question Four

Question

Show that the following vectors are linearly dependent:

$\vec{u}=\left[\begin{array}{c}-1\\ 1\end{array}\right],\vec{v}=\left[\begin{array}{c}\frac{1}{3}\\ -\frac{1}{3}\end{array}\right]$

Solution

Let $c_1,c_2\in \mathbb{R}$ satisfying $c_1\mathbf{u}+c_2\mathbf{v}=0⟹$

$$\left[\begin{array}{c}-c_1+\frac{1}{3}{c}_2\\ c_1-\frac{1}{3}{c}_2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$$

Here we can see the system of equations is unsolvable and has infinite solutions, as $\mathbf{u}$ and $\mathbf{v}$ are multiples of one another.

Question Five

Question

Determine whether the vector $\vec{u}=\left[\begin{array}{c}1\\ 2\end{array}\right]$ is a linear combination of $\vec{v}=\left[\begin{array}{c}0\\ 2\end{array}\right]$ and $\vec{w}=\left[\begin{array}{c}1\\ 4\end{array}\right]$.

If so, find the scalars.

Solution

If $\mathbf{u}$ is a linear combination of $\mathbf{v}$ and $\mathbf{w}$, then $\mathbf{u}=c_1\mathbf{v}+c_2\mathbf{w}⟹$

$$\left[\begin{array}{c}0+c_2\\ c_1+4c_2\end{array}\right]=\left[\begin{array}{c}1\\ 2\end{array}\right]⟹c_1=-2,c_2=1$$

Therefore, $\mathbf{u}$ is a linear combination of $\mathbf{v}$ and $\mathbf{w}$, where the scalars are $-2$ and $1$, respectively.