assume that T is a linear transformation Find the standard m

assume that T is a linear transformation .Find the standard matrix of T ?

In assume that T is a linear transformation. Find the standard matrix of T. T : R^2 rightarrow R^4 T(e_1) = (3,1,3,1)and T(e_2) =(-5,2,0,0), where e_1 = (1,0) and e_2 = (0,1).

Solution

1) The vector e1 is 2X1 vector and the resulting vector is a 4X1 vector, hence the T2 must be of the dimension 4X2 vector

let the vector T be

multiplying with e1, we get

a(1) + 0(b) = 3 => a = 3

c(1) + 0(d) = 3 => c = 1

e(1) + 0(f) = 3 => e = 3

g(1) + 0(h) = 3 => g = 1

multiplying with e2 we get

a(0) + 1(b) = 3 => b = -5

c(0) + 1(d) = 3 => d = 2

e(0) + 1(f) = 3 => f = 0

g(0) + 1(h) = 3 => h = 0

Hence the matrix T will be after substituting these values we get

a	b
c	d
e	f
g	h