Question

Let T:R

3
→R
3
be a linear transformation such that T(1,0,0)=(2,−1,4),T(0,1,0)=(1,−2,3), and T(0,0,1)=(2,−2,0). Find the indicated image.
T(1,2,−4)
T(1,2,−4)=
​

Answer 1

T(1,2,-4) = (-2, 3, 10).

Explanation:

To find T(1,2,-4), we can use the linearity of T. Since T is a linear transformation, we have:

T(a1,a2,a3) = a1T(1,0,0) + a2T(0,1,0) + a3T(0,0,1)

= a1(2,-1,4) + a2(1,-2,3) + a3(2,-2,0)

= (2a1 + a2 + 2a3, -a1 - 2a2 - 2a3, 4a1 + 3a2)

In this case, a1 = 1, a2 = 2, a3 = -4. Plugging these values into the expression above, we get:

T(1,2,−4) = (2(1) + 2(2) + 2(-4), -(1) - 2(2) - 2(-4), 4(1) + 3(2))

= (2 + 4 - 8, -1 - 4 + 8, 4 + 6)

= (-2, 3, 10)

Therefore, T(1,2,-4) = (-2, 3, 10).