Checking my answer : For which vectors the system has solution?

[fluidistic]

Homework Statement

I must describe implicitly the set of the vectors (b_1, b_2, b_3) such that the system has a solution.
x-y+2z+w=b_1
2x+2y+z-w=b_2
3x+y+3z=b_3.

2. The attempt at a solution
Playing with matrices I got that 0=b_3-b_1-b_2, so that I think that the system has solution for all b_1, b_2 and b_3 such that b_1=b_3-b_2.
Am I right? Also, did I describe the set of vectors implicitly? Ah maybe I should write (b_3-b_2, b_2, b_3), for all b_2 and b_3.

 

[mighty2000]

Hello,

Based on your attempt at a solution, it seems like you are on the right track. To describe the set of vectors implicitly, you can write it as {(b_3-b_2, b_2, b_3) | b_2, b_3 ∈ ℝ}, which means that for any real values of b_2 and b_3, the vector (b_3-b_2, b_2, b_3) will satisfy the given system of equations. This set of vectors can also be written as the set of all linear combinations of the vectors (1, 0, 1) and (-1, 1, 0), since these two vectors span the solution space.

However, it is important to note that this is not the only solution set for the given system of equations. There may be other sets of vectors that can also satisfy the system. To ensure that you have found the complete solution set, you can try solving the system using different methods, such as Gaussian elimination or matrix inversion.

I hope this helps clarify your understanding. Keep up the good work!