Answer:
(1) No matter what's the value of 
, 
is never in the span of 
and 
.
(2) The three vectors , , and are always linearly dependent for all real .
Step-by-step explanation:
<h3>(a)</h3>
If is in the span of and , there need to exist real 
and 
such that
.
Assume that such and 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]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20a%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%7B1%20%5C%5C%20-4%5C%5C2%7D%20%5Cend%7Barray%7D%5Cright%5D%20%2B%20b%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%7B-4%20%5C%5C%2016%5C%5C-8%7D%5Cend%7Barray%7D%5Cright%5D%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%20%5C%5C7%20%5C%5C%20h%5Cend%7Barray%7D%5Cright%5D)
.
![\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]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%7Ba%20%5C%5C%20-4a%5C%5C2a%7D%20%5Cend%7Barray%7D%5Cright%5D%20%2B%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%7B-4b%20%5C%5C%2016b%5C%5C-8b%7D%5Cend%7Barray%7D%5Cright%5D%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%20%5C%5C7%20%5C%5C%20h%5Cend%7Barray%7D%5Cright%5D)
.
.
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]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bcc%7Cc%7D%201%20%26%20-4%20%26%205%20%5C%5C%20-4%20%26%2016%20%26%207%20%5C%5C%202%20%26%20-8%20%26%20h%5Cend%7Barray%7D%5Cright%5D)
.
(Add four times row one to row two and 
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]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csim%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bcc%7Cc%7D%201%20%26%20-4%20%26%205%20%5C%5C%200%20%26%200%20%26%2027%20%5C%5C%200%20%26%200%20%26%20h%20-%2010%5Cend%7Barray%7D%5Cright%5D)
.
Note that in row two,
- Left-hand side:
; - Right-hand side:
.
In other words, this system is inconsistent. There's no real and that would satisfy the condition
.
Hence 
.
There's no real that allows , to be part of the span of and .
<h3>(b)</h3>
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
. In other words, can be written as the linear combination of the other two vectors. Additionally, since the coefficient in front of is zero, neither the exact value of nor the value of will make a difference. Therefore, for all 
, the three vectors , , and are linearly dependent.
