Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there any algorithm to determine whether a matrix is a linear combination of another set of matrices? For example, I want to find whether the matrix [1 0] can be written as a linear combination of the matrices [1 2] and [1 1].

share|cite|improve this question
I think I'll need to represent the linear combination of these matrices as a system of linear equations, and then convert the system of equations to row echelon form - how can I write this problem as a system of linear equations? – Anderson Green Dec 4 '12 at 5:27
up vote 1 down vote accepted

Solve the system of equations $\begin{pmatrix} a+b \\ 2a +b \end{pmatrix}=\begin{pmatrix} 1 \\ 0 \end{pmatrix}$. If there is a solution, then it can be written as a linear combination. If there is no solution, then it can't be.

share|cite|improve this answer
There must be some way to generalize this to matrices of any dimension. How would you find the solution if each of the matrices were 2x3 (instead of 2x1?) – Anderson Green Dec 4 '12 at 5:42
In general, given a matrix $A$ and a matrix $B$ and a matrix $C$, $C$ is a linear combination of $A$ and $B$ if there exists $x,y \in \mathbb{R}$ so that $xA+yB=C$. So solve $(xa_{ij}+ya_{ij})=(c_{ij})$. – Jebruho Dec 4 '12 at 5:46
In this case, what does aij, yaij and cij represent? – Anderson Green Dec 11 '12 at 6:02
$xa_{ij}$ is $x$ times the $i,j$ entry of $A$. $ya_{ij}$ is defined similarly. – Jebruho Dec 11 '12 at 20:12

Your Answer


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.