How to determine whether a set spans in Rn

[Idealistic]

In general, I'd like to know how to determine whether a set of m vectors spans in Rn. Do I have to look at the rank of the matrix that the vectors form?

Like if 3 vectors in R3, have a rank of 3, does this mean they span in R3?

 

[galactus]

A set of vectors spans if they can be expressed as linear combinations. Say we have a set of vectors we can call S in some vector space we can call V. The subspace, we can call W, that consists of all linear combinations of the vectors in S is called the spanning space and we say the vectors span W.

Here is an example of vectors in R^3.

Say we have $\displaystyle V_{1}=(1,1,2), \;\ v_{2}=(1,0,1), \;\ v_{3}=(2,1,3)$

We want to see if they span or not.

We have to find whether an arbitrary vector, say, $\displaystyle b=(b_{1},b_{2},b_{3})$ can be expressed as a linear combo $\displaystyle b=k_{1}v_{1}+k_{2}v_{2}+k_{3}v_{3}$ of the vectors $\displaystyle v_{1},v_{2},v_{3}$.

Set up a system of equations in terms of the components:

$\displaystyle (b_{1},b_{2},b_{3})=k_{1}(1,1,2)+k_{2}(1,0,1)+k_{3}(2,1,3)$

$\displaystyle (b_{1},b_{2},b_{3})=(k_{1}+k_{2}+2k_{3}, \;\ k_{1}+k_{3}, \;\ 2k_{1}+k_{2}+3k_{3})$

$\displaystyle k_{1}+k_{2}+2k_{3}=b_{1}$

$\displaystyle k_{1} \;\ \;\ +k_{3}=b_{2}$

$\displaystyle 2k_{1}+k_{2}+3k_{3}=b_{3}$

The system is consistent for all $\displaystyle b_{1},b_{2},b_{3}$ iff the matrix of coefficients:

$\displaystyle A=\begin{bmatrix}1&1&2\\1&0&1\\2&1&3\end{bmatrix}$

has a determinant that is not equal to 0.

But this determinant does equal 0, so it DOES NOT span.

 

[Idealistic]

Thank you for the clarification.

 

[DrMike]

In general, I'd like to know how to determine whether a set of m vectors spans in Rn. Do I have to look at the rank of the matrix that the vectors form?

Like if 3 vectors in R3, have a rank of 3, does this mean they span in R3?

I wouldn't want to say you had to look at the rank, but that will certainly do.

The columns - or rows - of a rank r matrix will span an r-dimensional space. If r=3 and the vectors are in R^3, then this must be the whole space.

However, that's not the only way to do it. For example, you could look at the null space, and use the rank-nullity theorem.