Determine which of the following mappings F are linear I'm having a really hard time understanding how to figure out if a mapping is linear or not. Here is my homework question:
Determine which of the following mappings F are linear.
(a) $F: \mathbb{R}^3 \to \mathbb{R}^2$ defined by $F(x,y,z) = (x, z)$
(b) $F: \mathbb{R}^4 \to \mathbb{R}^4$ defined by $F(X) = -X$
(c) $F: \mathbb{R}^3 \to \mathbb{R}^3$ defined by $F(X) = X + (0, -1, 0)$
Sorry about my formatting. I'm not sure how to write exponents and the arrow showing that the mapping is from R^n to R^m. Any help is greatly appreciated!!

To check if a mapping is linear in general, all you need is verify the two properties.


*

*$f(x+y) = f(x) + f(y)$

*$f(\alpha x) = \alpha f(x)$


The above two can be combined into one property: $f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$
Edit
For instance, if we want to show say $F(x) = f(x_1,x_2,x_3) = x_1 - 4x_2 + x_3$ is linear, where $x = (x_1,x_2,x_3)$, then all you need to do is as follows.
\begin{align*}
F(\alpha x + \beta y) & = f(\alpha x_1 + \beta y_1,\alpha x_2 + \beta y_2,\alpha x_3 + \beta y_3) \\
 & = (\alpha x_1 + \beta y_1) - 4(\alpha x_2 + \beta y_2) + (\alpha x_3 + \beta y_3)\\
 & = (\alpha x_1 - 4 \alpha x_2 + \alpha x_3) + (\beta y_1 - 4 \beta y_2 + \beta y_3)\\
 & = \alpha (x_1 - 4 x_2 + x_3) + \beta (y_1 - 4 y_2 + y_3)\\
 & = \alpha f(x_1,x_2,x_3) + \beta f(y_1,y_2,y_3)\\
 & = \alpha F(x) + \beta F(y)\\
\end{align*}
Hence the above function is linear.
EDIT
As Arturo points out problem $c$ is not a linear map because of the constant hanging around. Such maps are called affine maps. Affine maps are those for which $f(x) - f(0)$ is a linear map.
