mattgondek4

2021-11-07

Willie

Step1

$a{\mathbf{v}}_{1}+b{\mathbf{v}}_{2}$
$\text{for some numbers a and b.The numbers a and b are called the weights of the vector relative to}$
$\left\{{\mathbf{v}}_{1},{\mathbf{v}}_{2}\right\}$
Step2
So to get any vector in the span, we just pick any two numbers a and b form the linear combination with those weights.
Step 3
Vector1:Here is the vector in the span with weights 1 and 2.
$1\left[\begin{array}{c}7\\ 1\\ -6\end{array}\right]+2\left[\begin{array}{c}-5\\ 3\\ 0\end{array}\right]=\left[\begin{array}{c}7\\ 1\\ -6\end{array}\right]+\left[\begin{array}{c}-10\\ 6\\ 0\end{array}\right]=\left[\begin{array}{c}-3\\ 7\\ -6\end{array}\right]$
Vector2:Weights:0 and -2.
$0\left[\begin{array}{c}7\\ 1\\ -6\end{array}\right]-2\left[\begin{array}{c}-5\\ 3\\ 0\end{array}\right]=\left[\begin{array}{c}10\\ -6\\ 0\end{array}\right]$
Vector3:Weights:0 and 0.
$0\left[\begin{array}{c}7\\ 1\\ -6\end{array}\right]-2\left[\begin{array}{c}-5\\ 3\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
Vector4:Weights:5 and 7.
$5\left[\begin{array}{c}7\\ 1\\ -6\end{array}\right]+7\left[\begin{array}{c}-5\\ 3\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 26\\ -30\end{array}\right]$
Vector5:Weights:-3 and 1
$-3\left[\begin{array}{c}7\\ 1\\ -6\end{array}\right]+1\left[\begin{array}{c}-5\\ 3\\ 0\end{array}\right]=\left[\begin{array}{c}-26\\ 0\\ 18\end{array}\right]$.
Result:
$\left[\begin{array}{c}-4\\ 7\\ -6\end{array}\right],\left[\begin{array}{c}10\\ -6\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 26\\ -30\end{array}\right],\left[\begin{array}{c}-26\\ 0\\ 18\end{array}\right]$

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