Examples

## Ex 1

$\beta_1 =${$\vec{b}_1,\vec{b}_2$} where $\vec{b} = \begin{bmatrix} 2\\-1 \end{bmatrix} \text{ and } \vec{b}_2 = \begin{bmatrix} 0\\2 \end{bmatrix}$

rewrite the vector $\begin{bmatrix} \vec{x} \end{bmatrix}_{\beta_1} = \begin{bmatrix} 3\\1 \end{bmatrix}$ in the standard basis.

HINT: The standard basis can be written as the identity matrix in the proper dimension.

(1)
\begin{align} \vec{x}_{standard basis} = 3\begin{bmatrix} 2\\-1 \end{bmatrix} + 1\begin{bmatrix} 0\\2 \end{bmatrix} = \begin{bmatrix} 6\\-1 \end{bmatrix} \end{align}
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