1. ## Matrix transformations

What matrices represent these transformations with respect to the standard basis?
T: R^3 -----> R^3

1) T rotates each point through an angle theta about the x-axis in the direction that moves the point (0,0,1) towards the point (0,-1,0)

2) T reflects each point in the (x,z) - plane

2. Originally Posted by maximus101
What matrices represent these transformations with respect to the standard basis?
T: R^3 -----> R^3

1) T rotates each point through an angle theta about the x-axis in the direction that moves the point (0,0,1) towards the point (0,-1,0)

2) T reflects each point in the (x,z) - plane
Remember that you just need to find the transformation of the basis vectors in $\mathbb{R}^3$

For 1) this is just a counterclockwise rotation in the y,z plane. It will fix any point on the x axis. Now just draw a 2d picture to Find out what
$T(e_2)$ and $T(e_3)$ are.
Hint: the transform will not change the length of the vector it will stay on the unit circle Now just convert from polar back to rectangular coordinates.

3. To find the matrix representation of a linear transformation in a given basis, apply the transformation to each basis vector in turn and write the result as a linear combination of the basis vectors. The coefficients give the columns of the matrix.

For example, the "reflection in the xz-plane" maps each point (x,y,z) into (x,-y,z) so maps the basis vectors (x, 0, 0)-> (x, 0, 0), (0, y, 0)-> (0 -y, 0), (0, 0, z)-> (0, 0, z).