Section 1.5

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2. (a) Dimension of the domain space $= 2$ ; dimension of the range space $= 3$ ; $f$ is linear

(b) Dimension of the domain space $= 2$ ; dimension of the range space $= 2$ , $f$ is neither linear nor affine

(d) Dimension of the domain space $= 3$ ; dimension of the range space $= 2$ ; $f$ is linear

(e) Dimension of the domain space $= 3$ ; dimension of the range space $= 4$ ; $f$ is affine

(f) Dimension of the domain space $= 2$ ; dimension of the range space $= 1$ ; $f$ is affine

(g) Dimension of the domain space $= 1$ ; dimension of the range space $= 2$ ; $f$ is linear

(h) Dimension of the domain space $= 4$ ; dimension of the range space $= 2$ ; $f$ is linear

(i) Dimension of the domain space $= 2$ ; dimension of the range space $= 2$ ; $f$ is neither linear nor affine

(j) Dimension of the domain space $= 2$ ; dimension of the range space $= 3$ ; $f$ is neither linear nor affine

3. (a) $\displaystyle{M = \begin{bmatrix}1 & \phantom{-}1\\ 2 & -3\end{bmatrix}}$

(b) $\displaystyle{M = \begin{bmatrix}2 & 1 & -1 & \phantom{-}3\\ 1 & 2 & \phantom{-}0 & -3 \end{bmatrix}}$

(d) $\displaystyle{M = \begin{bmatrix}-5\end{bmatrix}}$

(e) $\displaystyle{M = \begin{bmatrix}4 & -3 & 2\end{bmatrix}}$

(f) $\displaystyle{M = \begin{bmatrix}1 & \phantom{-}1 & 1\\ 3 & -1 & 0\\ 0 & \phantom{-}1 & 2\end{bmatrix}}$

(g) $\displaystyle{M = \begin{bmatrix}2 & \phantom{-}0\\ 0 & \phantom{-}3\\ 1 & \phantom{-}1\\ 1 & -1\\ 2 & -3\end{bmatrix}}$

(h) $\displaystyle{M = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}}$

(i) $\displaystyle{M = \begin{bmatrix}2 & 1 & -1 & \phantom{-}3\\ 1 & 2 & \phantom{-}0 & -3\end{bmatrix}}$

5. (a) $\displaystyle{\begin{bmatrix}-4 \\ \phantom{-}4\end{bmatrix}}$

(b) $\displaystyle{\begin{bmatrix}-5\\ 11\\ -4\end{bmatrix}}$

(d) $\displaystyle{\begin{bmatrix}2 \\ 10\\ 3\end{bmatrix}}$

7. $\displaystyle{M = \begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}}$

8. $\displaystyle{M = \begin{bmatrix}\phantom{-}0 & -1\\ -1 & \phantom{-}0\end{bmatrix}}$

10. $\displaystyle{M = \begin{bmatrix}\phantom{-}\cos(\theta) & \phantom{-}\sin(\theta)\\ -\sin(\theta) & \phantom{-}\cos(\theta)\end{bmatrix}}$

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The Calculus of Functions of Several Variables

Section 1.5

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