Question Waiting for answers​​​​​​​ Let $$G: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$$ be the linear transformation defined by $G((1,0))=(1,0,2), \quad G((0,1))=(5,1,3),$ and let $$F: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$$ be the linear transformation defined by $F((u, v, w))=(u+v, u-w) .$ (a) Determine a formula for the composition $$F \circ G$$ explicitly (i.e. without using standard matrices so you must first express $$G$$ as a formula). (b) Compute the standard matrix for $$F \circ G$$ by finding those for $$F$$ and $$G$$, and compare your results.

Transcribed Image Text: Let $$G: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$$ be the linear transformation defined by $G((1,0))=(1,0,2), \quad G((0,1))=(5,1,3),$ and let $$F: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$$ be the linear transformation defined by $F((u, v, w))=(u+v, u-w) .$ (a) Determine a formula for the composition $$F \circ G$$ explicitly (i.e. without using standard matrices so you must first express $$G$$ as a formula). (b) Compute the standard matrix for $$F \circ G$$ by finding those for $$F$$ and $$G$$, and compare your results.
Transcribed Image Text: Let $$G: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$$ be the linear transformation defined by $G((1,0))=(1,0,2), \quad G((0,1))=(5,1,3),$ and let $$F: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$$ be the linear transformation defined by $F((u, v, w))=(u+v, u-w) .$ (a) Determine a formula for the composition $$F \circ G$$ explicitly (i.e. without using standard matrices so you must first express $$G$$ as a formula). (b) Compute the standard matrix for $$F \circ G$$ by finding those for $$F$$ and $$G$$, and compare your results.