## The Rank of the Sum of Two Matrices

## Problem 441

Let $A$ and $B$ be $m\times n$ matrices.

Prove that

\[\rk(A+B) \leq \rk(A)+\rk(B).\]

Let $A$ and $B$ be $m\times n$ matrices.

Prove that

\[\rk(A+B) \leq \rk(A)+\rk(B).\]

**(a)** Let $A=\begin{bmatrix}

1 & 3 & 0 & 0 \\

1 &3 & 1 & 2 \\

1 & 3 & 1 & 2

\end{bmatrix}$.

Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$.

**(b)** Find the rank and nullity of the matrix $A$ in part (a).

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 $A$.

Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix}

x_1 \\

x_2

\end{bmatrix}\right )=\begin{bmatrix}

x_1-x_2 \\

x_1+x_2 \\

x_2

\end{bmatrix}$.

**(a) **Show that $T$ is a linear transformation.

**(b)** Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for each $\mathbf{x} \in \R^2$.

**(c)** Describe the null space (kernel) and the range of $T$ and give the rank and the nullity of $T$.

Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.

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