Suppose that the vectors
\[\mathbf{v}_1=\begin{bmatrix}
-2 \\
1 \\
0 \\
0 \\
0
\end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix}
-4 \\
0 \\
-3 \\
-2 \\
1
\end{bmatrix}\]
are a basis vectors for the null space of a $4\times 5$ matrix $A$. Find a vector $\mathbf{x}$ such that
\[\mathbf{x}\neq0, \quad \mathbf{x}\neq \mathbf{v}_1, \quad \mathbf{x}\neq \mathbf{v}_2,\]
and
\[A\mathbf{x}=\mathbf{0}.\]

(Stanford University, Linear Algebra Exam Problem)

We are asked to find a vector $\mathbf{x}$ in the null space of $A$, which are not $\mathbf{0}, \mathbf{v}_1, \mathbf{v}_2$.

Recall that the null space is a vector space. Thus, any linear combination of vectors in the null space is still in the null space.

Since $\mathbf{v}_1, \mathbf{v}_2$ are basis of the null space of $A$, they are in particular vectors in the null space of $A$.
Thus, for example,
\[\mathbf{x}=2\mathbf{v}_2=\begin{bmatrix}
-4 \\
1 \\
0 \\
0 \\
0
\end{bmatrix}\]
is an element of the null space and it is not equal to $\mathbf{0}, \mathbf{v}_1, \mathbf{v}_2$.

Another example is
\[\mathbf{x}=\mathbf{v}_1+\mathbf{v_2}=\begin{bmatrix}
-6 \\
1 \\
-3 \\
-2 \\
1
\end{bmatrix}.\]

In general, you can prove that any vector of the form
\[\mathbf{x}=c_1\mathbf{v}_1+c_2\mathbf{v}_2,\]
where $c_1, c_2$ are scalars such that $(c_1, c_2)\neq (0,0), (1,0), (0, 1)$, satisfied the required conditions.

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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.
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(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 […]

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Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.
Define the map $f:\R^2 \to \calF[0, 2\pi]$ by
\[\left(\, f\left(\, \begin{bmatrix}
\alpha \\
\beta
\end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta […]

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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 […]

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Suppose that $n\times n$ matrices $A$ and $B$ are similar.
Then show that the nullity of $A$ is equal to the nullity of $B$.
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Let $A$ be an $m \times n$ matrix.
Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$.
Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$.
Then find $A\mathbf{w}$.
Hint.
Recall that the null space of an […]

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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 […]