Assume that T is a linear transformation Find the standard m

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) T: R^3 rightarrow R^2, T(e_1) = (1, 3), T(e_2) = (4, -7), and T(e_3) = (-5,4), where e_1, e_2, e_3 are the columns of the 3 Times 3 identity matrix. T : R^2 rightarrow R^2 rotates points (about the origin) through 3pi/2 radians (counterclockwise). T : R^2 rightarrow R^2 rotates points (about the origin) through -pi/4 radians (clockwise). T : R^2 rightarrow R^2 is a vertical shear transformation that maps e_1 into e_1 - 2e_2 but leaves the vector e_2 unchanged.

Solution

Post multiple problems to get the remaining answers. Thanks

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