Chapter 1, Problem 7SE

Consider the problem of determining whether the following system of equations is consistent for all b₁, b₂, b₃:

2x₁ – 4x₂     – 2x₃ = b₁

–5x₁ + x₂     + x₃     = b₂

7x₁ –  5x₂     – 3x₃ = 3

1. a. Define appropriate vectors, and restate the problem in terms of Span {v₁, v₂, v₃}. Then solve that problem.
2. b. Define an appropriate matrix, and restate the problem using the phrase “columns of A.”
3. c. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.