Question: a) Define what it means for a function \(F:{{\mathbb{R}}^{m}}\to {{\mathbb{R}}^{n}}\) to be a linear transformation.

b) Are the following linear transformations? Why? Note that the why part of the question is very important.

b1) \(F & :{{\mathbb{R}}^{2}}\to {{\mathbb{R}}^{2}},\,\,\left( \begin{matrix} x \\ y \\ \end{matrix} \right)\to \left( \begin{matrix} x+y-1 \\ 3x-y \\ \end{matrix} \right)\)

b2) \(F & :\mathbb{R}\to \mathbb{R},\,\,x\to \sin \left( x \right)\)

b3) \(F & :{{\mathbb{R}}^{3}}\to \mathbb{R},\,\,\vec{v}\to \left( -1,2,3 \right)\cdot \vec{v}\)

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