Linear Transformations

Question 1

formula for the projection of x on to a non zero vector u

Answer 1

(v.x)/(u.u) u

Question 2

proj[v] (x + y)

Answer 2

proj[v] (x) + proj[v] (y)

Question 3

proj[v] (rx)

Answer 3

rproj[v] (x)

Question 4

what is an orthogonal projection to the x-axis

Answer 4

a function that maps a vector U = [u1 u2] to [u1 0]

Question 5

projection matrix formula

Answer 5

Pu = 1/(|u|^2) (uuT)

Question 6

square of Pu

Answer 6

Pu

Question 7

what is the function that rotates vectors x radian the anticlockwise direction

Answer 7

[cosx - sinx]
[sinx cosx]

Question 8

how to prove a function is linear

Answer 8

apply the function to:
- f(x + y) and prove it = f(x) + f(y)
- f(cx) and prove it = cf(x)