Question

Use the following vectors to answer parts​ (a) and​ (b). v1equals=[Start 3 By 1 Matrix 1st Row 1st Column 1 2nd Row 1st Column negative 4 3rd Row 1st Column 2 EndMatrix ]1 −4 2 ​, v2equals=[Start 3 By 1 Matrix 1st Row 1st Column negative 4 2nd Row 1st Column 16 3rd Row 1st Column negative 8 EndMatrix ]−4 16 −8 ​, v3equals=[Start 3 By 1 Matrix 1st Row 1st Column 5 2nd Row 1st Column 7 3rd Row 1st Column h EndMatrix ]5 7 h ​(a) For what values of h is v3 in ​Span{v1​, v2​}? ​(b) For what values of h is ​{v1​, v2​, v3​} linearly​ dependent?

Answer 1

Answer:

(1) No matter what's the value of $h$, $\vec{v}_3$ is never in the span of $\vec{v}_1$ and $\vec{v}_2$.

(2) The three vectors $\vec{v}_1$, $\vec{v}_2$, and $\vec{v}_3$ are always linearly dependent for all real $h$.

Step-by-step explanation:

(a)

If $\vec{v}_3$ is in the span of $\vec{v}_1$ and $\vec{v}_2$, there need to exist real $a$ and $b$ such that

$a\; \vec{v}_1 + b\; \vec{v}_2 = \vec{v}_3$.

Assume that such $a$ and $b$ do exist.

In other words,

$\displaystyle a \left[\begin{array}{c}{1 \\ -4\\2} \end{array}\right] + b\left[\begin{array}{c}{-4 \\ 16\\-8}\end{array}\right] = \left[\begin{array}{c}5 \\7 \\ h\end{array}\right]$.

$\displaystyle \left[\begin{array}{c}{a \\ -4a\\2a} \end{array}\right] + \left[\begin{array}{c}{-4b \\ 16b\\-8b}\end{array}\right] = \left[\begin{array}{c}5 \\7 \\ h\end{array}\right]$.

$\left\{\begin{array}{rrcr} a & - 4b &=& 5\\ -4a & + 16b &= &7\\2a & -8b & =&h\end{aligned}\right.$.

Rewrite as an augmented matrix and row-reduce:

$\displaystyle \left[ \begin{array}{cc|c} 1 & -4 & 5 \\ -4 & 16 & 7 \\ 2 & -8 & h\end{array}\right]$.

(Add four times row one to row two and $-2$ times row one to row three.)

$\displaystyle \sim \left[ \begin{array}{cc|c} 1 & -4 & 5 \\ 0 & 0 & 27 \\ 0 & 0 & h - 10\end{array}\right]$.

Note that in row two,

• Left-hand side: $0$;
• Right-hand side: $27\neq 0$.

In other words, this system is inconsistent. There's no real $a$ and $b$ that would satisfy the condition

$a\; \vec{v}_1 + b\; \vec{v}_2 = \vec{v}_3$.

Hence $\forall h \in \mathbb{R}, \quad \vec{v}_3\not \in \text{Span}\{\vec{v}_1, \vec{v}_2\}$.

There's no real $h$ that allows $h$, $\vec{v}_3$ to be part of the span of $\vec{v}_1$ and $\vec{v}_2$.

(b)

If the three vectors are linearly dependent, at least one of them can be expressed as the linear combination of the other two.

Note that

$\vec{v}_2 = (-4)\vec{v}_1 + 0 \; \vec{v}_3$. In other words, $\vec{v}_2$ can be written as the linear combination of the other two vectors. Additionally, since the coefficient in front of $\vec{v}_3$ is zero, neither the exact value of $\vec{v}_3$ nor the value of $h$ will make a difference. Therefore, for all $h \in \mathbb{R}$, the three vectors $\vec{v}_1$, $\vec{v}_2$, and $\vec{v}_3$ are linearly dependent.