In which $\R^k$, are the Nullspace and Range Subspaces?

Problem 712

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 $A$.

Solution.

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 $1$.

More from my site

• 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 […]
• Matrix Representation, Rank, and Nullity of a Linear Transformation $T:\R^2\to \R^3$ Let $T:\R^2 \to \R^3$ be a linear transformation such that \[T\left(\, \begin{bmatrix} 3 \\ 2 \end{bmatrix} \,\right) =\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \text{ and } T\left(\, \begin{bmatrix} 4\\ 3 \end{bmatrix} […]

You may also like...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Prove Vector Space Properties Using Vector Space Axioms

Using the axiom of a vector space, prove the following properties. Let $V$ be a vector space over $\R$. Let...

Close