Intersection of Two Null Spaces is Contained in Null Space of Sum of Two Matrices
Problem 311
Let $A$ and $B$ be $n\times n$ matrices. Then prove that
\[\calN(A)\cap \calN(B) \subset \calN(A+B),\]
where $\calN(A)$ is the null space (kernel) of the matrix $A$.
Recall that the null space (or kernel) of an $n \times n$ matrix is
\[\calN(A)=\{\mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}\}.\]
The null space $\cal(N)$ is a subspace of the $n$-dimensional vector space $\R^n$.
Proof.
Let $\mathbf{x}$ be an arbitrary vector in the intersection $\calN(A)\cap \calN(B)$.
Then the vector $\mathbf{x}$ belongs to both $\calN(A)$ and $\calN(B)$.
Thus, by definition of the null space, we have
\[A\mathbf{x}=\mathbf{0} \text{ and } B\mathbf{x}=\mathbf{0}.\]
If follows from these equalities that we have
\begin{align*}
(A+B)\mathbf{x}=A\mathbf{x}+B\mathbf{x}=\mathbf{0}+\mathbf{0}=\mathbf{0}.
\end{align*}
Hence $\mathbf{x}$ lies in the null space of the matrix $A+B$, that is, $\mathbf{x}\in \calN(A+B)$.
Since $\mathbf{x}$ is an arbitrary element of $\calN(A)\cap \calN(B)$, we have shown the inclusion
\[\calN(A)\cap \calN(B) \subset \calN(A+B),\]
as required.
The Null Space (the Kernel) of a Matrix is a Subspace of $\R^n$
Let $A$ be an $m \times n$ real matrix. Then the null space $\calN(A)$ of $A$ is defined by
\[ \calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.\]
That is, the null space is the set of solutions to the homogeneous system $A\mathbf{x}=\mathbf{0}_m$.
Prove that the […]
Dimension of Null Spaces of Similar Matrices are the Same
Suppose that $n\times n$ matrices $A$ and $B$ are similar.
Then show that the nullity of $A$ is equal to the nullity of $B$.
In other words, the dimension of the null space (kernel) $\calN(A)$ of $A$ is the same as the dimension of the null space $\calN(B)$ of […]
Find a Basis For the Null Space of a Given $2\times 3$ Matrix
Let
\[A=\begin{bmatrix}
1 & 1 & 0 \\
1 &1 &0
\end{bmatrix}\]
be a matrix.
Find a basis of the null space of the matrix $A$.
(Remark: a null space is also called a kernel.)
Solution.
The null space $\calN(A)$ of the matrix $A$ is by […]
Prove a Given Subset is a Subspace and Find a Basis and Dimension
Let
\[A=\begin{bmatrix}
4 & 1\\
3& 2
\end{bmatrix}\]
and consider the following subset $V$ of the 2-dimensional vector space $\R^2$.
\[V=\{\mathbf{x}\in \R^2 \mid A\mathbf{x}=5\mathbf{x}\}.\]
(a) Prove that the subset $V$ is a subspace of $\R^2$.
(b) Find a basis for […]
Row Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a Matrix
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 […]
Determine Null Spaces of Two Matrices
Let
\[A=\begin{bmatrix}
1 & 2 & 2 \\
2 &3 &2 \\
-1 & -3 & -4
\end{bmatrix} \text{ and }
B=\begin{bmatrix}
1 & 2 & 2 \\
2 &3 &2 \\
5 & 3 & 3
\end{bmatrix}.\]
Determine the null spaces of matrices $A$ and $B$.
Proof.
The null space of the […]
Quiz 6. Determine Vectors in Null Space, Range / Find a Basis of Null Space
(a) Let $A=\begin{bmatrix}
1 & 2 & 1 \\
3 &6 &4
\end{bmatrix}$ and let
\[\mathbf{a}=\begin{bmatrix}
-3 \\
1 \\
1
\end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix}
-2 \\
1 \\
0
\end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix}
1 \\
1 […]