Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm self-studying a bit of introductory linear algebra, watching the lectures on MIT OCW given by Gilbert Strang. The course isn't too rigourous and gives many things without proof, most of which I can reason through and convince myself of so far, but I've run into one thing that I can't wrap my head around.

So he's discussing an algorithm to solve the system of linear equations $\boldsymbol{\mathbf{A}}\boldsymbol{\mathbf{x}} = \boldsymbol{\mathbf{b}}$. And basically he says that all we need to do is find a particular solution to the equation (after elimination) and then add it to any vector in the nullspace of the matrix. It's obvious why the sum of the particular solution and the nullspace vector is part of the solution set to $\boldsymbol{\mathbf{A}}\boldsymbol{\mathbf{x}} = \boldsymbol{\mathbf{b}}$. It's not obvious to me, however, why all solutions to the equation can be described as that type of sum. Could someone explain to me why this is true?

share|improve this question
2  
If $\mathbf{Ax} = \mathbf{b} = \mathbf{Ay}$ then $\mathbf{A}(\mathbf{x}-\mathbf{y}) = \mathbf{Ax} - \mathbf{Ay} = \mathbf{b}-\mathbf{b}= \mathbf{0}$, so $\mathbf{A}(\mathbf{x}-\mathbf{y}) = 0$ and hence $\mathbf{x}-\mathbf{y}$ is an element of the nullspace of $\mathbf{A}$. In words: any two solutions $\mathbf{x}$ and $\mathbf{y}$ differ by an element of the nullspace of $\mathbf{A}$. –  t.b. Jul 12 '11 at 16:00
    
And this is analogous to the recipe to find the general solution of a linear differential equation: the "particular" solution plus the "general" solution of the homogeneous equation. –  leonbloy Jul 12 '11 at 16:23
add comment

1 Answer

up vote 3 down vote accepted

Sure, if $Ax_1 = b$ and $Ax_2 = b$ then by linearity, $A(x_1 - x_2) = 0$ hence $x_1-x_2$ is a nullspace vector.

share|improve this answer
1  
Of course, can't believe I missed that, thanks @Gortaur –  Arpon Jul 12 '11 at 16:03
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.