# ML Wiki

## 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 $R$