Question

What is the relationship between the solution space of Ax = b and the solution set of the corresponding homogenous system Ax = 0?

Answer 1

Like if you know that Ax = 0 has a unique solution, is the same true for Ax = b?

Answer 2

Translations added.

Answer 3

The idea is that if you a solution for the null space of homogenous system, the solution for the corresponding nonhomogenous system plus any vector from the null space of the corresponding homogenous system is a solution.

Answer 4

the column space of A and the Null space of A are orthogonal subspaces. So if x_1 is a vector that satisfies Ax=b, and x_2 is a vector that satisfies Ax = 0, x_1 and x_2 are orthogonal.

Answer 5

Oh okay I understand thanks forgot about the orthogonal part

Answer 6

Hmm...I would have said they were parallel.

Answer 7

Here's a picture i drew answering one other Linear Algebra problem. the matrix is a projection matrix onto the plane with vectors that have y = x (in R^3) The Column space is the plane I drew, the null space is the line orthogonal to it.

Answer 8

The solutions of Ax = b are x +V and x is a particular solution, V is the solutions of Ax=0.

Answer 9

Yes this is what I was talking about. This is by linearity, consider L_A(x + v) where x is a solution and v \in N(L_A) then L_A(x + v) = L_A(x) + L_(v) = L_A(x) + 0 = b

Answer 10

The shift is orthogonal, the solution space is parallel.

Answer 11

An affine plane in R3 is just a translation away from the origin.

Answer 12

i stand corrected, this is what i was thinking of:

Answer 13

i switched the row space of A and the column space of A in that definition, my bad >.<