Basic Linear Algebra - Linear Transformations Help

**Lord Voldemort** · December 7th 2010, 02:48 PM

So I know that a transformation is linear if T(u+v) = T(u) + T(v) and T(kv) = kT(v).
I'm given a system
y1 = b - c
y2 = ac
y3 = a - b
and to be honest I have no idea where to start.
Any help is appreciated.

**dwsmith** · December 7th 2010, 02:56 PM

Are you trying to determine if $\displaystyle y_1, \ y_2, \ \mbox{and} \ y_3$ are each a linear transformation?

**Lord Voldemort** · December 7th 2010, 03:03 PM

No, I think it's referring to the system as a whole. Maybe interpreted as a matrix? I'm not quite sure. My book is somewhat vague on this topic so I really quite confused.

EDIT: The answer in the back of the book says non-linear, so yes, I'd assume it's referring if the whole taken as a transformation is linear.

**dwsmith** · December 7th 2010, 03:23 PM

$\displaystyle \begin{bmatrix} b-c\\ ac\\ a-b \end{bmatrix},\ \begin{bmatrix} y-z\\ xz& \\ x-y & \end{bmatrix}\in S$

$\displaystyle \begin{bmatrix} b-c\\ ac\\ a-b \end{bmatrix} +\begin{bmatrix} y-z\\ xz& \\ x-y & \end{bmatrix}=\begin{bmatrix} b-c +y-z\\ ac+xz& \\ a-b+x-y & \end{bmatrix}=\begin{bmatrix} (b+y)-(c+z)\\ ac+xz& \\ (a+x)-(b+y) & \end{bmatrix}\neq \begin{bmatrix} (b+y)-(c+z)\\ (a+x)(c+z)& \\ (a+x)-(b+y) & \end{bmatrix}$

**Lord Voldemort** · December 7th 2010, 03:38 PM

Thank you very much!
For some reason it never occurred to me to just pick another set of points to prove/disprove it.
Now, I also have the system
y1 = 2b, y2 = 3c, y1 = a ,
Am I correct in saying that this is linear b/c
2b + 2y = 2(b+y), 3(c + z) = 3c + 3z, and so forth?

Also, considering the system
y1 = 2x, y2 = x + 2, y3 = 2x, I'm confused again...
y1 = y3 clearly are linear, and I'm a little confused about proving how y2 is/isn't.

**dwsmith** · December 7th 2010, 03:49 PM

Originally Posted by Lord Voldemort

Thank you very much!
For some reason it never occurred to me to just pick another set of points to prove/disprove it.
Now, I also have the system
y1 = 2b, y2 = 3c, y1 = a ,
Am I correct in saying that this is linear b/c
2b + 2y = 2(b+y), 3(c + z) = 3c + 3z, and so forth?

Also, considering the system
y1 = 2x, y2 = x + 2, y3 = 2x, I'm confused again...
y1 = y3 clearly are linear, and I'm a little confused about proving how y2 is/isn't.

Yeah correct for the first one. You also need to show scalar multiplication.

$\displaystyle (x+2)+(\beta+2)=x+\beta+4\neq (x+\beta)+2$

Equations of the form y = mx + b are affine transformations.

**Lord Voldemort** · December 7th 2010, 03:59 PM

I'm not quite sure what an affine transformation is yet, is it just a nonlinear one?
(In regards to your last explanation)
Basically, y2 adds two to the the value input, but because adding two to both values and then adding the values as compared to adding the values as one value in and of itself and then adding two to that results in an inequality, it's not linear?
I guess I didn't really know how to approach the issue.

Thanks so much, I was a bit unclear on this but I definitely get this now!

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