Finding the Transformation Matrix if it's linear

[silvermane]

Homework Statement

Which of the transformations are linear? If they are, then find the transformation matrix.
the input is v = (v1,v2)

a. t(v) = (v2,v1)
b. t(v) = (v1,v2)
c. t(v) = (0,v1)
d. t(v) = (0,1)

The Attempt at a Solution

a. it is linear
b. it is linear
c. I think it is linear because we're going from R2 to R1 and not going to the origin.
d. I don't think this one is linear, because all values can't map to the same point.

Now, to find the transformation matrix A, I would like a helpful tip/hint/procedure for how to find the matrix A. I am having some difficulty understanding this concept, as it's not in the book and this part was added by the teacher.

Thank you for your help in advance! :)
(I'm off to bed, but will be back in the morning, hehe)

 

[╔(σ_σ)╝]

Your linear stuff is correct.

The first one is simply a row operation of switching rows.

The second one is an identity map.

The third is a projection.

How about the last one ? It is not linear. Can you find a transformation matrix ? Remember every matrix transformation is linear.

All these a standard transformations with their matrices in most books or google. You should be able to come up with them one your known .

You can find matrices such that when you multiply your column vector by them you get the desired result. It shouldn't be too difficult.

Going to bed too.:-)

 

[D H]

d. t(v) = (0,1)

d. I don't think this one is linear, because all values can't map to the same point.

The result is correct, but your reasoning is not. t(v) = (0,0) is (trivially) linear, and it maps everything to the same point.

 

[╔(σ_σ)╝]

*Awaken*

The man is right; pershaps, you should show that the maps fails one of the test of linearity.