General question about matrices

[mooshupork34]

I know that if a vector is in the row space of a given matrix, then it means that:

vector = c1[row 1] + c2[row 2] + cn[row n]

Then one must put the matrix in row reduced echelon form in order to find the c values.

However, what does one do if one wants to determine if a vector (say [3, 1, 4]) is in the column space of a 3 by 2 matrix?

Let's say the 3 by 2 matrix is:

[1 4
2 0
2 1]

 

[daon]

The vector is in the column space of a matrix iff it can be formed by a linear combination of the columns. In that case, find a basis for the column space and see if you can form the vector.

Your particular example depends also on the entries (i.e. are they from a field like the Reals? or the Integers? etc).

 

[JakeD]

...what does one do if one wants to determine if a vector (say [3, 1, 4]) is in the column space of a 3 by 2 matrix? Let's say the 3 by 2 matrix is:

[1 4
2 0
2 1]

You have to solve Ax = b, where A is the 3x2 matrix and b = [3,1,4], or determine that no solution exists. For a solution to exist, the rref of the augmented matrix [A|b] must have all zeros in the bottom row and for the top two rows either ones on the diagonal or all zeros in the row.

EDIT: Corrected typo in augmented matrix.

 

[mooshupork34]

You have to solve Ax = b, where A is the 3x2 matrix and b = [3,1,4], or determine that no solution exists. For a solution to exist, the rref of the augmented matrix [A|c] must have all zeros in the bottom row and for the top two rows either ones on the diagonal or all zeros in the row.

Thanks! So is it correct if I set up the matrix to be put in RREF like so?:

1 4 | 3
2 0 | 1
2 1 | 4

 

[JakeD]

So is it correct if I set up the matrix to be put in RREF like so?:

1 4 | 3
2 0 | 1
2 1 | 4

Yes. There was a typo in my augmented matrix, which I fixed.