Explained: "What is the span of a matrix?"

College
Algebra
Linear algebra
Vectors and spaces

Alisha Pitts

Answered question

2022-01-22

What is the span of a matrix?

Answer & Explanation

Darian Sosa

Beginner2022-01-23Added 12 answers

Explanation:
A set of vectors spans a space if every other vector in the space can be written as a linear combination of the spanning set. But to get to the meaning of this we need to look at the matrix as made of column vectors.
Here's an example in

R^{2}

:
Let our matrix

M = (\begin{array}{cc} 1 & 2 \\ 3 & 5 \end{array})

This has column vectors:

(\begin{matrix} 1 \\ 3 \end{matrix})

and

(\begin{matrix} 2 \\ 5 \end{matrix})

which are linearly independent, so the matrix is non-singular ie invertible etc etc.
Let's say that we want to show that the generalised point (x,y) is within the span of these 2 vectors, ie so that the matrix spans all of

R^{2}

, then we look to solve this:

α (\begin{matrix} 1 \\ 3 \end{matrix}) + β (\begin{matrix} 2 \\ 5 \end{matrix}) = (\begin{matrix} x \\ y \end{matrix})

(\begin{array}{cc} 1 & 2 \\ 3 & 5 \end{array}) (\begin{matrix} α \\ β \end{matrix}) = (\begin{matrix} x \\ y \end{matrix})

You can solve this is any number of ways, eg row reduce or invert M to get:

α = - 5 x + 2 y, β = 3 x - y

So let's say we want to check that (2,3) is in the span of this matrix, M, we apply the result we just got:

α = - 4

β = 3

Double-check:

- 4 (\begin{matrix} 1 \\ 3 \end{matrix}) + 3 (\begin{matrix} 2 \\ 5 \end{matrix}) = (\begin{matrix} 2 \\ 3 \end{matrix})

Consider next a different matrix:

M^{'} = (\begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array})

. This is singular because its column vectors,

(\begin{matrix} 1 \\ 2 \end{matrix})

and

(\begin{matrix} 2 \\ 4 \end{matrix})

, are linearly dependent. This matrix only spans along direction

(\begin{matrix} 1 \\ 2 \end{matrix})

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Linear algebra

Didn't find what you were looking for?