Question

Let T: R2 R2 be a linear transformation such that T((1, 2)) (2, 3) and T((0, 1)) = (1, 4). Then T((5, =

-4)) is
1. (-4,-41)
2. (-6,1)
3. (-1,6)
4. (1,-6)

Answer 1

To find the linear transformation T((5, -4)), we express (5, -4) as a linear combination of the given vectors, and apply linearity to obtain T((5, -4)) = (-4, -41).

Step-by-step explanation:

The linear transformation T is defined by its action on two basis vectors of R2. We are given T((1, 2)) = (2, 3) and T((0, 1)) = (1, 4). To find T((5, -4)), we use the property of linearity of the transformation, which allows us to express (5, -4) as a linear combination of the two given vectors. First, find the coefficients a and b such that (5, -4) = a*(1, 2) + b*(0, 1). We get the system of equations:

1a + 0b = 5

2a + 1b = -4

Solving for a and b gives us a=5 and b=-14. Then, apply the transformation T using the linearity property to get:

• T((5, -4)) = T(5*(1, 2) - 14*(0, 1))
• = 5*T((1, 2)) + (-14)*T((0, 1))
• = 5*(2, 3) + (-14)*(1, 4)
• = (10, 15) + (-14, -56)
• = (-4, -41)

Therefore, the transformed vector T((5, -4)) is (-4, -41), which corresponds to option 1.