Domain, Co-Domain, and Linearity of Linear Systems homework check.

I am asked to find the domain, co-domain, and to determine whether of not the transformation is linear. I'm not sure if I am doing this properly, so I figured I would ask as my textbook doesn't have solutions.

If someone could explain how to determine the linearity of the system I would appreciate it, right now I am just guessing.

$$w_1 = 3x_1 - 2x_2 +4x_3$$

$$w_2 = 5x_1 - 8x_2 + x_3$$

Domain = $R^3$

Co-Domain = $R^2$

Linearity: False?

$$w_1 = 2x_1x_2 - x_2$$

$$w_2 = x_1 + 3x_1x_2$$

$$w_3 = x_1 + x_2$$

Domain = $R^2$

Co-Domain = $R^3$

Linearity: False?

$$w_1 = 5x_1 - x_2 + x_3$$

$$w_2 = -x_1 + x_2 + 7x_3$$

$$w_3 = 2x_1 - 4x_2 - x_3$$

Domain: $R^3$

Co-Domain: $R^3$

Linearity: False?

$$w_1 = x_1^2-3x_2+x_3-2x_4$$

$$w_2 = 3x_1 - 4x_2 -x_3^2 + x_4$$

Domain: $R^4$

Co-Domain: $R^2$

Linearity: False?

For all of these, you have the codomain and domain correct, but some of them are in fact linear. Remember that a function $f: \Bbb{R}^n \rightarrow \Bbb{R}^m$ is defined to be linear if the following two conditions hold:

1. For all $u$, $v$ in $\Bbb{R}^n$, $f(u+v) = f(u) + f(v)$
2. For all $u$ in $\Bbb{R}^n$ and $a$ in $\Bbb{R}$, $f(a \cdot u) = af(u)$

This does hold for your first and third functions (for example, because, in the first function, letting $f(x_1,x_2,x_3) = (w_1,w_2)$, $$f(x_1+y_1,x_2+y_2,x_3+y_3) = \\(3x_1 + 3y_1 - 2x_1 - 2y_1+4x_1 + 4y_1, 5x_1 + 5y_1 - 8x_1 - 8y_1 + x_1 + y_1) = \\(3x_1 - 2x_1 + 4x_1, 5x_1 - 8x_1 + x_1) + (3y_1 - 2y_1 + 4y_1, 5y_1 - 8y_1 + y_1) = \\f(x_1,x_2,x_2) + f(y_1,y_2,y_3),$$ and essentially the same argument gives the second property. You should think about what, exactly, it is about the second and fourth equations that makes them nonlinear.

OK, looks like you have the Domain and Co-Domain sorted.

A function is Linear if it is

• Additive - $f(X+Y)=f(X)+f(Y)$ for all $X,Y$, and
• Homogeneous of order 1 - $f(\alpha X)=\alpha f(X)$ for all $\alpha$.

Some of these are, some are not.