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?
$\endgroup$2 Answers
$\begingroup$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:
- For all $u$, $v$ in $\Bbb{R}^n$, $f(u+v) = f(u) + f(v)$
- 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.
$\endgroup$ $\begingroup$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.
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