T R3 rightarrow R2 Te1 13 Te2 4 7 and Te3 5 4 where e1 e

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.

Solution

The vector e1,e2 and e3 are all column vectors of size 3X1

The resulting vector is of size 2X1, hence the size of T matrix must be (2X3) since if we multiply 2X3 with 3X1 we will yield a 2X1 vector

Let the T matrix be of order 2X3 with the given unknowns

multiplying T with e1 we get

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

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

multiplying T with e2 we get

a(0) + b(1) + c(0) = 4 => (b=4)

d(0) + e(1) + f(0) = -7 => (e=-7)

multiplying T with e3 we get

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

d(0) + e(0) + f(1) = 4 => (f=4)

Hence the final T matrix will be

a	b	c
d	e	f