Basic question, vectors into matrices

[Shawj02]

Hi,

My question is concerned with vectors, null space, solving for independence and subspaces.

Now, I've found If a question gives a set of vectors and asks find if they are independent or not The vectors are written in columns (to make a matrix) and then Gaussian elimination is applied.
But in questions like solving for a null space (Ax=0) or subspaces the vectors are written across rows.

Yeah, The notes from my lectures are a bit funny, they seem to switch between writing vectors along columns or rows quiet regularly. If anyone has some general rules about when to write vectors either way, I'd be very grateful.

(If anyone tries to explain something with a diagram/matrix I'd prefer it to be done with 3 vectors in 4 dimensions. Sorry for being fussy but I think because this is a question about the uncertainty of columns/row I'd think it would be best.)

Thanks!

 

[matt grime]

But in questions like solving for a null space (Ax=0) or subspaces the vectors are written across rows

Which vectors are written across rows?

The rule is it doesn't matter which way you write things (rows or columns) as long as you only do things that make sense (row operations or column operations).


If I wanted to find a nice basis for the span of some vectors, such as reduced echelon form, then I would write the vectors as rows and do row operations, and the resulting rows would be what I was after.

If all you want to find is the rank, then it is immaterial since row rank and column rank are the same.


If I wanted to know if a vector, b, was in the span of others, then I can either write as rows and try to solve

xM=b (b written as a row vector)

or I can write as columns and try to solve

Nx=b (b written as a column vector)

You just need to think what xM means and Nx means. The former combines rows of M, the latter combines columns of N.