How to expand the span of a vector set in linear algebra?

[wishnight]

Homework Statement

I do understand that in matrix 3x2, the set of vector doesn't span of R3. What should I do to make the set of vector span of R3.

Homework Equations

The Attempt at a Solution

I think adding one more set of vector is the best idea. So, if I can add one more set of vector to make them span of R3, how to find the third set of vector. thanks

 

[conquest]

I think you need to state the question more clearly. In a 3x2 matrix the columns don't span R^3. The three standard vectors (1,0,0), (0,1,0), (0,0,1) span R^3. I you want to make a matrix of which the colums span R^3 put these in a matrix.

If you want to use the two vectors you already got from the matrix:
- first check to see if the two you have are linearly independent.
i.e. whether one is a mulitple of the other (in the case of 2 vectors)
- If they are find a third vector that is also linearly independt of both the vectors (together)
- I they aren't find two more that, together with one of the vectors you already had,
are linearly independent of each other.

Then put them in a matrix.

 

[wishnight]

I mean if we got 2 vector a = (1,2,1) and b = (1,3,1). Then I need to create one more vector that will make the set of vectors span of R3

 

[conquest]

There are many way's to do this. One way is to put these to vectors and (a,b,c) in a matrix. Calculate the determinant then choose a,b,c such that the determinant is not 0. Any such a,b,c will do