Linear algebra question

Question

Use the function to find the image of v and the preimage of w.

T(v₁, v₂, v₃) = (4v₂ − v₁, 4v₁ + 5v₂), v = (3, −4, −1), w = (6, 18)

Answer 1

The definition of T illustrates that the linear transformation takes inputs from R3 and produces outputs in R2, meaning it is a matrix A with 2 rows and 3 columns.

[A₁₁ A₁₂ A₁₃] [4v₂ - v₁ ]

[A₂₁ A₂₂ A₂₃] * v = [4v₁ + 5v₂]

But we also know from standard multiplication that

[A₁₁ A₁₂ A₁₃] [A₁₁v₁ + A₁₂v₂ + A₁₃v₃ ]

[A₂₁ A₂₂ A₂₃] * v = [A₂₁v₁ + A₂₂v₂ + A₂₃v₃]

We can match the coefficients for v to determine the matrix entries:

[-1 4 0]

[4 5 0] = A

The image of v is the product Av, and the preimage of w is asking to solve for the unknown input v that leads to Av=w (which you can solve for using row reduction).

IMAGE

Av = [-1(3)+4(4)+0(-1); 4(3)+5(-4)+0(-1)] = [13, -8]

PREIMAGE

Form the augmented matrix [A \ w] and get to rref.

[-1,4,0,6]

[4,5,0,18]

Add 4xRow1 to Row2

[-1,4,0,6]

[0,21,0,42]

Divide rows so pivots are 1

[1,-4,0,-6]

[0,1,0,2]

Add 4xRow2 to Row1

[1,0,0,2]

[0,1,0,2]

This tells us we need a preimage vector of the form v = [2, 2, c], where c is any constant.