The Rank of the Sum of Two Matrices
Problem 441
Let $A$ and $B$ be $m\times n$ matrices.
Prove that
\[\rk(A+B) \leq \rk(A)+\rk(B).\]
Add to solve later Tagged: column space
Add to solve later Quiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix
Problem 320
(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).
Add to solve later 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 $A$.
Add to solve later Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$
Problem 154
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 $T$.
Add to solve later Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)
Problem 136
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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