Find the general form of the span of th e indicated matrices, span(A1, A2) B=begin{bmatrix}2 & 5 0 & 3 end{bmatrix} , A_1=begin{bmatrix}1 & 2 -1 & 1 end{bmatrix} , A_2=begin{bmatrix}0 & 1 2 & 1 end{bmatrix}

Question

Find the general form of the span of th e indicated matrices, span(A1, A2)

B = [\begin{matrix} 2 & 5 \\ 0 & 3 \end{matrix}], A_{1} = [\begin{matrix} 1 & 2 \\ - 1 & 1 \end{matrix}], A_{2} = [\begin{matrix} 0 & 1 \\ 2 & 1 \end{matrix}]

Answer 1

Step 1
We have to find the general form of the span of the indicated matrices, span(A1, A2) $A_{1} = [\begin{matrix} 1 & 2 \\ - 1 & 1 \end{matrix}], A_{2} = [\begin{matrix} 0 & 1 \\ 2 & 1 \end{matrix}]$
Step 2
$(A_{1}, A_{2}) = c_{1} [\begin{matrix} 1 & 2 \\ - 1 & 1 \end{matrix}] + c_{2} [\begin{matrix} 0 & 1 \\ 2 & 1 \end{matrix}]$
$= [\begin{matrix} c_{1} & 2 c_{1} + c_{2} \\ - c_{1} + 2 c_{2} & c_{1} + c_{2} \end{matrix}]$
$= [\begin{matrix} w & x \\ y & z \end{matrix}]$
$[\begin{matrix} 1 & 0 & | w \\ 2 & 1 & | x \\ - 1 & 2 & | y \\ 1 & 1 & | z \end{matrix}] = [\begin{matrix} 1 & 0 & w \\ 1 & 0 & x - z \\ 0 & 2 & y + w \\ 1 & 1 & z \end{matrix}]$
$= [\begin{matrix} 1 & 0 & w \\ 1 & 0 & x - z \\ 0 & 1 & \frac{y + w}{2} \\ 1 & 1 & z - w \end{matrix}]$
Step 3
$= [\begin{matrix} 1 & 0 & w \\ 0 & 0 & x - z - w \\ 0 & 1 & \frac{y + w}{2} \\ 0 & 0 & z - w - \frac{y + w}{2} \end{matrix}]$
$w = x - z$
$z - w = \frac{y + w}{2}$
$2 (z - w) = y + w$
$2 z - 3 w = y$
$2 z - 3 (x - z) = y$
$5 z - 3 x = y$
$(A_{1}, A_{2}) = [\begin{matrix} x - z & x \\ 5 z - 3 x & z \end{matrix}]$
Step 4
$(A_{1}, A_{2}) = [\begin{matrix} x - z & x \\ 5 z - 3 x & z \end{matrix}]$