I am in a University Linear Algebra course and am confused by the term span and its relation to both matrices and vectors. Can someone help clarify what they mean?
=Span=
Can it only be made of vectors (1 row matrix)? (or can you "solve a span" of large matrices? $3\times 5, 4\times 9$, etc.)
In reduced echelon form will each vector/matrix act the same inside that span? (as an example, my professor says suppose matrix $A$ in $\mathbb{R}^4$ has the property that $Ax = b$ gives one unique solution, but how can each $B$-value act the same? Esp. if you can create inconsistent solutions.
With $\mathbb{R}^3$ or $\mathbb{R}^n$ does that mean this is just a concept of a span containing ALL vectors with $N$ columns(or is it rows?).
If you have an n-space, you need at least n vectors to "fill" that n-space. And that's true only if the vectors are linearly independent.
(If any two vectors are linearly dependent, they "collapse" into one, and you have only n-1 vectors that don't quite fill the n-space.) In row reduced echelon form, that means you have one (or more) rows will all zeroes.
If you have the n independent (e.g. column) vectors, they can be put into a matrix with dimension n (by definition).