Find the Pre-Image

Problem

x*([a],[b],[d])=[[5*a+3*b−2*d],[4*b−d]]=[[14],[2]]

Solution

1. Set up a system of linear equations by equating the components of the transformation output to the target vector.

5*a+3*b−2*d=14

4*b−d=2

2. Solve the second equation for d in terms of b to express the variables using a parameter.

d=4*b−2

3. Substitute the expression for d into the first equation to solve for a in terms of b

5*a+3*b−2*(4*b−2)=14

5*a+3*b−8*b+4=14

5*a−5*b=10

a−b=2

a=b+2

4. Express the general solution vector by letting b be a free parameter t

a=t+2

b=t

d=4*t−2

5. Write the pre-image as a set of vectors or in vector form.

[[a],[b],[d]]=[[t+2],[t],[4*t−2]]

[[a],[b],[d]]=[[2],[0],[−2]]+t*[[1],[1],[4]]

Final Answer

x(−1)*([14],[2])=[[t+2],[t],[4*t−2]],t∈ℝ

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