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

Vector Space Problems and Solutions

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 $A$.

 
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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 $1$.


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