Question

given T(e1), T(e2), and v how do i find T(v)?

Answer 1

are T(e1) and T(e2) functiions?

Answer 2

T(e1) = (1, √3) and T(e2)= (-√3, 1)

Answer 3

e1 and e2 are standard vectors (1,0) and (0,1)

T is a transformation matrix with column vectors, say: u1 and u2

T(e1) = 1u1 + 0u2 = (1,sqrt(3))
T(e2) = 0u1 + 1u2 = (-sqrt(3),1)

I know theres a simple way to do it, but that method wasnt a strong point in linear algebra for me.

T= a c
b d

T(e1) = 1(a) + 0(c) = (1)
1(b) 0(d) = (sqrt(3))

hence a=1, b=sqrt(3)


T(e2) = 0(a) + 1(c) = (-sqrt(3))
0(b) 1(d) = (1)

hence c=-sqrt(3), d=1
\[T=\begin{pmatrix} 1&-\sqrt{3}\\\sqrt{3}&1 \end{pmatrix}\]

thats how i usually did it in class