Let $\R^2$ be the $x$-$y$-plane. Then $\R^2$ is a vector space. A line $\ell \subset \mathbb{R}^2$ with slope $m$ and $y$-intercept $b$ is defined by
\[ \ell = \{ (x, y) \in \mathbb{R}^2 \mid y = mx + b \} .\]

Prove that $\ell$ is a subspace of $\mathbb{R}^2$ if and only if $b = 0$.

We must prove two statements. First we show that if $b \neq 0$, then $\ell$ is not a subspace. Then we show that if $b=0$, then $\ell$ is a subspace.

In order for $\ell$ to be a subspace, it must contain the zero element $(0, 0)$. Plugging this point into the defining equation yields $b=0$. Thus if $b \neq 0$, then $\ell$ cannot be a subspace.

Now we prove that if $b=0$, then $\ell$ is a subspace. We have already shown that $(0, 0) \in \ell$. Now suppose we have two points $(x_1, y_1) , (x_2 , y_2) \in \ell$. Then we have
\[y_1 + y_2 = m x_1 + m x_2 = m(x_1 + x_2)\]
and so $(x_1 + x_2 , y_1 + y_2)$ is contained in $\ell$.
Finally for $c \in \mathbb{R}$ we must show that $c(x_1 , y_1) = ( cx_1 , c y_1 )$ lies in $\ell$. We check
\[c y_1 = c ( m x_1) = m (c x_1),\]
and so $(c x_1 , c y_1 ) \in \ell$. This proves that if $b=0$, then $\ell$ is a subspace.

All Linear Transformations that Take the Line $y=x$ to the Line $y=-x$
Determine all linear transformations of the $2$-dimensional $x$-$y$ plane $\R^2$ that take the line $y=x$ to the line $y=-x$.
Solution.
Let $T:\R^2 \to \R^2$ be a linear transformation that maps the line $y=x$ to the line $y=-x$.
Note that the linear […]

The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane
Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$.
Then find the matrix representation of the linear transformation $T$ with respect to the […]

12 Examples of Subsets that Are Not Subspaces of Vector Spaces
Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.
(1) \[S_1=\left \{\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\]
in […]

Subspace Spanned by Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$
Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$.
Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$.
(a) Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ […]

Hyperplane in $n$-Dimensional Space Through Origin is a Subspace
A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors
\[\begin{bmatrix}
x_1 \\
x_2 \\
\vdots \\
x_n
\end{bmatrix}\in \R^n\]
satisfying the linear equation of the form
\[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\]
[…]

Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$
Let $S$ be the following subset of the 3-dimensional vector space $\R^3$.
\[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \]
where $\Z$ is the set of all integers.
[…]

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 […]