2.2 Span

1. Determine whether or not the given vector $\mathbf{b}$ is in the span of the vectors ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$.

a. ${\mathbf{u}}_1=\left[\begin{array}{l}1\\ 2\end{array}\right],{\mathbf{u}}_2=\left[\begin{array}{c}-1\\ -3\end{array}\right],\mathbf{b}=\left[\begin{array}{c}4\\ -2\end{array}\right]$

We need to check if a linear combination of $u_1$ and $u_2$ can produce $b$, by creating an augmented matrix and reducing it:

$$\left[\begin{array}{cccc}1& -1& |& 4\\ 2& -3& |& -2\end{array}\right]\to \left[\begin{array}{cccc}1& 0& |& 14\\ 0& 1& |& 10\end{array}\right]$$

so it is in the span:

$$14u_1+10u_2=\left[\begin{array}{c}4\\ -2\end{array}\right]$$

b. ${\mathbf{u}}_1=\left[\begin{array}{c}1\\ 0\\ -2\end{array}\right],{\mathbf{u}}_2=\left[\begin{array}{l}0\\ 2\\ 1\end{array}\right],\mathbf{b}=\left[\begin{array}{c}1\\ -4\\ -4\end{array}\right]$

It is in the span:

$$1u_1-2u_2=\left[\begin{array}{c}1\\ -4\\ -4\end{array}\right]$$

c. ${\mathbf{u}}_1=\left[\begin{array}{c}1\\ 0\\ -2\end{array}\right],{\mathbf{u}}_2=\left[\begin{array}{l}0\\ 2\\ 1\end{array}\right],\mathbf{b}=\left[\begin{array}{c}4\\ -1\\ 4\end{array}\right]$
It is not in the span because the reduced augmented matrix is a unit matrix.

2. Find a vector in ${\mathbf{R}}^3$ that is not in span

$\left\{\left[\begin{array}{c}2\\ -1\\ 4\end{array}\right],\left[\begin{array}{c}0\\ 3\\ -2\end{array}\right]\right\}$.

$\left[\begin{array}{c}3\\ -1\\ 4\end{array}\right]$ Cannot be in the span

3. Determine if the following vectors span ${\mathbf{R}}^2$.

$${\mathbf{u}}_1=\left[\begin{array}{l}3\\ 2\end{array}\right],{\mathbf{u}}_2=\left[\begin{array}{l}5\\ 3\end{array}\right]$$

$u_2$ and $u_1$ are not scalar multiples of one another and so they are not parallel, and do span $R^2$

4. Determine if the following vectors span ${\mathbf{R}}^3$.

$${\mathbf{u}}_1=\left[\begin{array}{c}2\\ -2\\ 0\end{array}\right],{\mathbf{u}}_2=\left[\begin{array}{c}1\\ -1\\ 1\end{array}\right]$$

Any two vectors $R_3$ vectors form a plane, and cannot span all of $R_3$

5. Give an example of a set of vectors that span ${\mathbf{R}}^n$.

$$\left[\begin{array}{c}1\\ 0\\ ⋮\\ 0_n\end{array}\right],\left[\begin{array}{c}0\\ 1\\ ⋮\\ 0_n\end{array}\right]\dots \left[\begin{array}{c}0\\ 0\\ ⋮\\ 1_n\end{array}\right]$$

6. Find all values of $h$ such that the set of vectors $\left\{{\mathbf{u}}_1,{\mathbf{u}}_2,{\mathbf{u}}_3\right\}$ spans ${\mathbf{R}}^3$.

$${\mathbf{u}}_1=\left[\begin{array}{c}1\\ 2\\ -3\end{array}\right],{\mathbf{u}}_2=\left[\begin{array}{l}0\\ h\\ 1\end{array}\right],{\mathbf{u}}_3=\left[\begin{array}{c}-3\\ 2\\ 1\end{array}\right]$$

The determinant of an augmented matrix for this system will give us the volume of the span. So long as it is nonzero, that value of h will result in a system which spans $R_3$

$$A=\left[\begin{array}{cccc}1& 0& |& -3\\ 2& h& |& 2\\ -3& 1& |& 1\end{array}\right]$$ $$det(A)=-8h-8$$ $$h\ne -1\to span(u_1,u_2,u_3)=R^3$$

7. Find $A,\mathbf{x}$, and $\mathbf{b}$ such that the equation $A\mathbf{x}=\mathbf{b}$ corresponds to the linear system:

$$\begin{array}{rl}2x_1+3x_2-5x_4& =-1\\ 4x_1+7x_3+x_4& =4\\ 3x_2-x_3+2x_4& =0\end{array}$$ $$\left[\begin{array}{cccc}2& 3& 0& -5\\ 4& 0& 7& 1\\ 0& 3& -1& 2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{c}-1\\ 4\\ 0\end{array}\right]$$ $$x=\left[\begin{array}{c}5s-1.1\\ 0.4-1.66s\\ 1.2-3s\\ s\end{array}\right]$$

8. Determine if the equation $A\mathbf{x}=\mathbf{b}$ has a solution for every choice of $\mathbf{b}$.

The equation will have a solution for every choice of b if it spans it’s entire vector space.
a. $A=\left[\begin{array}{cc}1& 3\\ -2& -6\end{array}\right]$

$$det(A)=0$$

Span < $R^2$ so there is not a solution for every choice of b.

b. $A=\left[\begin{array}{ccc}1& 0& -2\\ 2& 1& 2\\ 2& 1& 3\end{array}\right]$

$$det(A)=1$$

Span = $r^3$ so there is a solution for every choice of b.