## Row Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a Matrix

## Problem 260

Let \[A=\begin{bmatrix}

1 & 1 & 2 \\

2 &2 &4 \\

2 & 3 & 5

\end{bmatrix}.\]

**(a)** Find a matrix $B$ in reduced row echelon form such that $B$ is row equivalent to the matrix $A$.

**(b)** Find a basis for the null space of $A$.

**(c)** Find a basis for the range of $A$ that consists of columns of $A$. For each columns, $A_j$ of $A$ that does not appear in the basis, express $A_j$ as a linear combination of the basis vectors.

**(d)** Exhibit a basis for the row space of $A$.