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

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

First, we will tackle the image of v under this transformation, T. The image is going to describe the set of all possible outputs for a given set of inputs under a given transformations. In layman's terms, we take all of the given inputs, and plug it into the function. Whatever we get out is our image.

For the transformation above, T(v₁,v₂,v₃) = (4v₂ − v₁, 4v₁ + 5v₂). To find the image of v, we plug v into T.

T(v)=T(3,-4,-1) = (4*(-4)-(3),4(3)+5(-4))

We then simplify to get our final answer

Im(v)=(-19,-8)

A preimage is the set of all inputs that result in a given input under a given transformations. In layman's terms, we will take our output (w) and see what all of the possible inputs are that result in our output.

Let P be our preimage...

P=(p₁,p₂,p₃)

T(P)=w=(6,18)

(6,18)=(4p₂-p₁,4p₁+5p₂)

Breaking this up into a system of two equations...

6=4p₂-p₁

18=4p₁+5p₂

We can now solve the system of equations for p₁ and p₂ to get the following...

p₁=2, p₂=2

p₃ does not appear anywhere in the system of equations, thus it is a free variable. This implies any vector of the form (2,2,p₃) will map to w under T. (T(2,2,p₃)=w for all possible p₃). Thus we can say the following is our preimage...

preimage(w)={(2,2,p₃) | p₃ is an element of all real numbers}