Linear Transformations with matrices and vectors?

The linear transformation f of R^2 is given by the matrix:


2 1


-1 2





u =


1


0





v =


0


1





w =


1


2





z =


-3


4


i) find the length of each of the following vectors and the angle which the vector makes with the positive x- axis





ii) If you were to draw a diagram showing all of the vectors, explain what the linear transformation is doing to the vectors:





a) u


b) f(u)


c) v


d) f(v)


e) w


f) f(w)


g) z


h) f(z)

Linear Transformations with matrices and vectors?
I will do z and f(z) and leave you to do the others.


length of vector z = sqrt((-3)^2 + 4^2) = sqrt (9 + 16) = 5


the matrix moves z to 2*(-3) + 1*4 = -6 + 4 = -2 over


(-1)*(-3) + 2*4 = 3 + 8 = 11 so f(z) = (-2, 11)


length of f(z) = sqrt((-2)^2 + 11^2) = sqrt(4 + 121) = sqrt(125)


Angles can be found by using tanA = y-value/x-value