# Definition of rank and its characterstics by column vectors with a 3x4 matrix

How do we know that?

## Definition of rank and its characterstics by column vectors with a 3x4 matrix

A square matrix is singular if and only if its determinant is 0. In the next video I'll prove to you why it works. For an r x c matrix, If r is less than c, then the maximum rank of the matrix is r. Let me just call that matrix R. And that's just with respect to each other. That's a pivot entry. Let's replace it with it plus 2 times the second row. And sometimes you kind of get a headache thinking about doing something like this, but this wasn't too bad. And all bases have the same number of vectors for any given subspace. So these guys can definitely be represented as linear combinations of these guys. The rank of a matrix is defined as a the maximum number of linearly independent column vectors in the matrix or b the maximum number of linearly independent row vectors in the matrix. Both definitions are equivalent. You're just going to get a bunch of 0's. If you did then, or I guess a better way to think it, you don't need them to span, although they are part of the span.

Or what is the dimension-- not the dimension of the basis-- what is the dimension of the column space of A? Jump to navigation Jump to search Possible form of a matrix In linear algebraa matrix is in echelon form if it has the shape resulting from a Gaussian elimination. So they're linearly independent.

These guys are also linearly independant, which I haven't proven. And this is not a pivot entry, because it's following obviously another.

And now we can answer another question. Take A, put it into reduced row echelon form, see which columns are pivot columns. And in order for them to span, obviously all of these 5 vectors, if you have all of them, that's going to span your column space by definition.

Now the next question is what is the dimension of the basis? Previously, we showed how to find the row echelon form for matrix A. This is a2, a3, a4.

Therefore, only row echelon forms are considered in the remainder of this article.

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