Each plane $P$ in $\R^3$ through the origin is given by the equation
\[ax+by+cz=0\]
for some real numbers $a, b, c$.
That is, the plane $P$ is a set of vectors $\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}$ satisfying the equation $ax+by+cz=0$:
\[P=\left\{\, \begin{bmatrix}
x \\
y \\
z
\end{bmatrix}\in \R^3 \quad \middle| \quad ax+by+cz=0 \,\right \}.\]

Now the equation can be written as the matrix equation
\[A\mathbf{x}=\mathbf{0},\]
where $A$ is the $1\times 3$ matrix $A$, $\mathbf{x}\in \R^3$, and $\mathbf{0}$ is the $1$-dimensional zero vector given by
\[A=\begin{bmatrix}
a & b & c \\
\end{bmatrix}, \mathbf{x}=\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}, \mathbf{0}=[0].\]

Thus, the plane can be written as
\begin{align*}
P&=\{\mathbf{x}\in \R^3 \mid A\mathbf{x}=\mathbf{0}\}\\
\end{align*}
and this is the definition of the nullspace $\calN(A)$ of $A$. That is
\[P=\calN(A).\]

Therefore, the plane $P$ is the nullspace of the $1\times 3$ matrix $A$.
Since the nullspace of a matrix is always a subspace, we conclude that the plane $P$ is a subspace of $\R^3$.
Therefore, every plane in $\R^3$ through the origin is a subspace of $\R^3$.

Linear Dependent/Independent Vectors of Polynomials
Let $p_1(x), p_2(x), p_3(x), p_4(x)$ be (real) polynomials of degree at most $3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent?
(a) At $1$ each of the polynomials has the value $0$. Namely $p_i(1)=0$ […]

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 kernel of $T$ is $n-1$.
(The kernel of $T$ is also called the null space of $T$.)
(b) Let […]

Linear Properties of Matrix Multiplication and the Null Space of a Matrix
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 […]

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 […]

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 […]

Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$
Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings.
(a) $\rk(AB) \leq \rk(A)$.
(b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$.
Hint.
The rank of an $m \times n$ matrix $M$ is the dimension of the range […]

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 […]