Dimension of 4x4 Matrix: Find Basis Vectors

[DryRun]

Homework Statement

$$A= \begin{bmatrix}1 & 1 & 1 & 1 \\ 2 & 1 & 0 & -1 \\ 3 & 4 & 5 & 6 \\ -1 &2 &1&0 \end{bmatrix}$$Determine the dimension of A and give a set of basis vectors for A.

Homework Equations

Dimension of matrix, ref form of matrix.

The Attempt at a Solution

I reduced the matrix to row echelon form and then the dimension = rank of matrix. Is that correct? I am quite confused about what dimension means. In a 4x4 matrix, maybe dimension is 16? or is it the number of non-zero matrix elements?

 

[sid9221]

Easy way to remember it is:

# of columns - # of non zero rows in rref

 

[DryRun]

So, the dimension of A is the rank of the matrix A?

 

[HallsofIvy]

Yes. In fact, I would consider the term "dimension of a matrix" very strange. The rank of a matrix is the dimension of the image of the matrix.

 

[sid9221]

I think by dimension you mean "nullity" cause our lecturer also used "dimension of a matrix" which was confusing when studying from other sources.

Try plugging your matrix in wolfram and ask for nullity.

In the output it states a "dimension" which is always exactly what I always needed. So maybe that's what it means.

 

[Ray Vickson]

Homework Statement

$$A= \begin{bmatrix}1 & 1 & 1 & 1 \\ 2 & 1 & 0 & -1 \\ 3 & 4 & 5 & 6 \\ -1 &2 &1&0 \end{bmatrix}$$Determine the dimension of A and give a set of basis vectors for A.

Homework Equations

Dimension of matrix, ref form of matrix.

The Attempt at a Solution

I reduced the matrix to row echelon form and then the dimension = rank of matrix. Is that correct? I am quite confused about what dimension means. In a 4x4 matrix, maybe dimension is 16? or is it the number of non-zero matrix elements?

How does your textbook or lecturer or course notes define the term "dimension of a matrix"?

RGV