Question

Given that\begin{bmatrix} -2 & -2 & 3 \\ 1 & 2 & -2 \\ -1 & -1 & 1 \end{bmatrix} \begin{bmatrix} 18 \\ -12 \\ 8 \end{bmatrix} \sim \begin{bmatrix} 1 & 0 & 0 & -4 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 2 \end{bmatrix}, \\ write \begin{bmatrix} 18 \\ -12 \\ 8 \end{bmatrix} as a linear combination of the vectors \begin{bmatrix} -2 \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 2 \\ -1 \end{bmatrix}, \begin{bmatrix} 3 \\ -2 \\ 1 \end{bmatrix}. \\ \begin{bmatrix} 18 \\ -12 \\ 8 \end{bmatrix} = \boxed{} \begin{bmatrix} -2 \\ 1 \\ -1 \end{bmatrix} + \boxed{} \begin{bmatrix} -2 \\ 2 \\ -1 \end{bmatrix} + \boxed{} \begin{bmatrix} 3 \\ -2 \\ 1 \end{bmatrix}.

          Given that\begin{bmatrix} -2 & -2 & 3 \\ 1 & 2 & -2 \\ -1 & -1 & 1 \end{bmatrix} \begin{bmatrix} 18 \\ -12 \\ 8 \end{bmatrix} \sim \begin{bmatrix} 1 & 0 & 0 & -4 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 2 \end{bmatrix}, \\ write \begin{bmatrix} 18 \\ -12 \\ 8 \end{bmatrix} as a linear combination of the vectors \begin{bmatrix} -2 \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 2 \\ -1 \end{bmatrix}, \begin{bmatrix} 3 \\ -2 \\ 1 \end{bmatrix}. \\ \begin{bmatrix} 18 \\ -12 \\ 8 \end{bmatrix} = \boxed{} \begin{bmatrix} -2 \\ 1 \\ -1 \end{bmatrix} + \boxed{} \begin{bmatrix} -2 \\ 2 \\ -1 \end{bmatrix} + \boxed{} \begin{bmatrix} 3 \\ -2 \\ 1 \end{bmatrix}.

Given that
< b m a t r i x >

< b m a t r i x >
∼
< b m a t r i x >
,
write
< b m a t r i x >
as a linear combination of the vectors
< b m a t r i x >
,
< b m a t r i x >
,
< b m a t r i x >
.

< b m a t r i x >
=
< b m a t r i x >
+
< b m a t r i x >
+
< b m a t r i x >
.

Added by Robin R.

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