Let $A$ be an $m \times n$ matrix.
Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$.

For an $m \times n$ matrix $A$, the nullspace consists of vectors $\mathbf{x}$ such that $A\mathbf{x}=\mathbf{0}$. Thus, such an $\mathbf{x}$ must be $n$-dimensional. Since the nullspace is a subspace in $\R^3$, we conclude that $n=3$.

The range of $A$ consists of vectors $\mathbf{y}$ such that $\mathbf{y}=A\mathbf{x}$ for some $\mathbf{x}\in \R^n$. Hence, $\mathbf{y}$ is $m$-dimensional. As the range is a subspace in $\R^5$, we conclude that $m=5$.

Since a plane is a $2$-dimensional subspace, the nullity of $A$ is $2$.

The range is spanned by a single vector $\mathbf{v}$. Thus, $\{\mathbf{v}\}$ is a basis for the range. Thus, the rank is $1$.

Here is another way to see this. By the rank-nullity theorem, we have
\[\text{rank of $A$ + nullity of $A = n$}.\]
Since $n=3$ and the nullity is $2$, the rank is $1$.

Determine Bases for Nullspaces $\calN(A)$ and $\calN(A^{T}A)$
Determine bases for $\calN(A)$ and $\calN(A^{T}A)$ when
\[
A=
\begin{bmatrix}
1 & 2 & 1 \\
1 & 1 & 3 \\
0 & 0 & 0
\end{bmatrix}
.
\]
Then, determine the ranks and nullities of the matrices $A$ and $A^{\trans}A$.
Solution.
We will first […]

Quiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix
(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).
Solution.
(a) […]

A Linear Transformation $T: U\to V$ cannot be Injective if $\dim(U) > \dim(V)$
Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$.
Consider a linear transformation $T:U\to V$.
Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one).
Hints.
You may use the folowing facts.
A linear […]

Idempotent Matrices are Diagonalizable
Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.
We give three proofs of this problem. The first one proves that $\R^n$ is a direct sum of eigenspaces of $A$, hence $A$ is diagonalizable.
The second proof proves […]

Rank and Nullity of a Matrix, Nullity of Transpose
Let $A$ be an $m\times n$ matrix. The nullspace of $A$ is denoted by $\calN(A)$.
The dimension of the nullspace of $A$ is called the nullity of $A$.
Prove the followings.
(a) $\calN(A)=\calN(A^{\trans}A)$.
(b) $\rk(A)=\rk(A^{\trans}A)$.
Hint.
For part (b), […]

Linear Transformation to 1-Dimensional Vector Space and Its Kernel
Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation.
Prove the followings.
(a) The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$.
(b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the […]

Orthonormal Basis of Null Space and Row Space
Let $A=\begin{bmatrix}
1 & 0 & 1 \\
0 &1 &0
\end{bmatrix}$.
(a) Find an orthonormal basis of the null space of $A$.
(b) Find the rank of $A$.
(c) Find an orthonormal basis of the row space of $A$.
(The Ohio State University, Linear Algebra Exam […]