## Row Space

This is one of the Four Fundamental Subspaces

A *Row Space* $C(A^T)$ of a matrix $A$ is all linear combinations of rows of $A$, or all combinations of columns of $A^T$

Row Space

- $\text{dim } C(A^T) = r = \text{dim } C(A)$, there are $r$ pivot rows - the same dim as for Column Space
- basis: maximal system of linearly independent vectors from $A^T$
- when we get Row Reduced Echelon Form $R$ by applying Gaussian Elimination to $A$, the column space changes, so $C(A) \ne C(R)$
- but because we did row operations the row space should remain the same: it changed only the column space
- so $C(A^T) = C(R^T)$
- alternatively, we can take first $r$ rows of $R$ for the basis

Why the row space remains the same?

- all the operations were performed on rows - and we allowed to do only linear combinations
- so each linear operation gives us rows from the same space
- we may change the basis, but the space remains the same
- and cleanest form of the row space is the rows from $R$

## See Also

## Sources