Linear Algebra: Matrix vs Span help.

Question

The concept of span and matrix are getting jumbled, and I would like to get some clarification before I fall too far off the cliff.

Suppose I have a 2x2 matrix (all rows/columns are independent). Then, this is a system of two lines (based on number of columns) in 2D ambient space (based on number of rows).

However, the span of the columns (two vectors) is a plane, since it's a linear combination of the two vectors.

Am I correct?

Answer 1

Three little points to correct:

1. A $2\times 2$ matrix by itself does not represent any lines/ planes/ etc. But the matrix equation $Ax=b$, where $x$ and $b$ are column matrices, does represent a system of two lines in the plane.
2. You have the "based on number of []" backwards.
3. The span of the columns will only be the entire plane if the columns are linearly independent. If they are linearly dependent, but not both zero, then they span a line. If you have the zero matrix, then the columns span a point (the origin).