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!!

1, Yes is a linear mapping, F(ax)=aF(x), F(x+y)=F(x)+F(y)
2, Yes is a linear mapping, F(ax)=aF(x), F(x+y)=-x-y=F(x)+F(y)
3, Not a linear mapping F(0) not equal 0 
