muckymotter

If anyone could explain how the following is done, it would be greatly appreciated!

Consider the set L of all linear transformations. Let L_1: V -> W and L_2: V -> W be linear transformations. Define a vector addition on linear transformations as (L_1 + L_2): V -> W, where (L_1 + L_2)(v) = L_1(v) + L_2(v). Define also a scalar multiplication on linear transformations as (c * L_1): V -> W, where (c * L_1)(v) = cL_1(v). Using these operations, we may consider L a vector space.

Here is the question:

Show that for every L_1 ε L, there exists some L_2 ε L such that L_1 + L_2 + 0_L.

Thanks! I'm a little confused, though -

The first matrix Q is a 3x3 matrix, but the transformation matrix R is a 4x3 so I can't multiply them.

Consider the quadrilateral (namely Q) in R^3 formed by the points

(1, 0, 0), (2, 0, 0), (1, 1, 3), and (2, 1, 3).

(a) What should the coordinates be for the figure R we get by rotating Q counterclockwise in the x-y plane by 45 degrees, then dilating it by a factor of 3/2, then translating it along the vector (-2, 1, -1)?

Okay, so what I did was I used the matrix

cos45 -sin45 0

sin45 cos45 0

0 0 1

After this, multiplied the identity matrix for R^3 by 3/2 and then multiplied it by the matrix with 45 substituted for theta.

Then T(x,y,z)=(-2+3sqrt(2)x/4-3sqrt(2)y/4,1+3sqrt(2)x/4+3sqrt(2)y/4,-1+3z/2)

I substituted each of the quadrilateral points in for T(x, y, z) to come up with the four points and got:

(3/(2sqrt2) - 2, 3/(2sqrt2) + 1, -1),

(3/sqrt2 - 2, 3/sqrt2 + 1, -1),

(-2, 3/sqrt2 + 1, 3.5),

(3/(2sqrt2) - 2, 9/(2sqrt2) + 1, 3.5)

But my trouble lies in doing (b) so if anyone could explain how it is done, I would be really grateful as I have been stuck on this for days!:

(b) What matrix transforms Q into R?