1. ## Matrices question

Given a 4x5 matrix, what could a row of 0's represent geometrically?

My attempt:

Given a scalar PLANE equation, you could make a 3x3 matrix and solve the system of equations.
A row of zeroes there could represent a consisten but dependent solution,
The planes would intersect in a line.
But PLANES only have 3 variables.
we were given FOUR equations and FIVE variables and we had to make a matrix out of that.

I'm not sure what shape that would make because first of all, whatever that equation represents, is in 4-space. It COULD be a moving plane, right?

But what would four of those make if they intersect at a line?

2. A 4 by 5 matrix, with 4 rows and 5 columns maps a vector in $\displaystyle R^4$ to $\displaystyle R^5$. The "range" of such a matrix, the subspace of $\displaystyle R^5$ into which all of $\displaystyle R^4$ is mapped, cannot have dimension larger than 4. If, in addition, one row is all "0"s (or the matrix can be row reduced to that), then the range cannot have dimension greater than 3. If, after row-reduction, that is the only row of all "0"s, the range has dimension 3. Of course, that would mean that the nullspace has dimension 5- 3= 2.

3. Like how HallsofIvy explained that given matrix must be in 3 dimension. So that excludes the possibility of 4 dimension. But, given the possibility of a 4 dimension, there could also be n dimensional matrices. It gets really hairy geometrically representing any dimension greater than 3. Its best to stick with algebraic and logical proofs of these greater dimensions. You're on the right path with this question, because you'll find higher level math courses that delve into this question, which should satisfy your curiosity given you keep vigilant with the current curriculum.