For what real values of $a$ is the set
\[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\]
a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?

The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by $\mathbf{0}(x) = 0$. This shows that $\mathbf{0} \in W_a$ if and only if $a=0$.

We have shown that if $a \neq 0$, then $W_a$ is not a subspace as every subspace contains the zero vector. Now we consider the case $a=0$ and prove that $W_0$ is a subspace.

We verify the subspace criteria: the zero vector of $C(\R)$ is in $W_0$, and $W_0$ is closed under addition and scalar multiplication.

As mentioned before, $\mathbf{0} \in W_0$.

Now suppose $f, g \in W_0$. Then $f(0) = g(0) = 0$, and so
\[(f+g)(0) = f(0) + g(0) = 0.\]
Thus $f+g \in W_0$. Finally, if $c \in \mathbb{R}$ is a scalar and $f \in W_0$, then
\[(cf)(0) = c f(0) = c \cdot 0 = 0.\]
Thus $cf \in W_0$, and $W_0$ is a vector subspace.

Subspaces of the Vector Space of All Real Valued Function on the Interval
Let $V$ be the vector space over $\R$ of all real valued functions defined on the interval $[0,1]$. Determine whether the following subsets of $V$ are subspaces or not.
(a) $S=\{f(x) \in V \mid f(0)=f(1)\}$.
(b) $T=\{f(x) \in V \mid […]

The Centralizer of a Matrix is a Subspace
Let $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define
\[W = \{ A \in V \mid AM = MA \}.\]
The set $W$ here is called the centralizer of $M$ in $V$.
Prove that $W$ is a subspace of $V$.
Proof.
First we check that the zero […]

The Sum of Subspaces is a Subspace of a Vector Space
Let $V$ be a vector space over a field $K$.
If $W_1$ and $W_2$ are subspaces of $V$, then prove that the subset
\[W_1+W_2:=\{\mathbf{x}+\mathbf{y} \mid \mathbf{x}\in W_1, \mathbf{y}\in W_2\}\]
is a subspace of the vector space $V$.
Proof.
We prove the […]

Prove that the Center of Matrices is a Subspace
Let $V$ be the vector space of $n \times n$ matrices with real coefficients, and define
\[ W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.\]
The set $W$ is called the center of $V$.
Prove that $W$ is a subspace […]

Sequences Satisfying Linear Recurrence Relation Form a Subspace
Let $V$ be a real vector space of all real sequences
\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \cdots).\]
Let $U$ be the subset of $V$ defined by
\[U=\{ (a_i)_{i=1}^{\infty} \in V \mid a_{k+2}-5a_{k+1}+3a_{k}=0, k=1, 2, \dots \}.\]
Prove that $U$ is a subspace of […]

Determine Whether a Set of Functions $f(x)$ such that $f(x)=f(1-x)$ is a Subspace
Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let
\[W=\{ f(x)\in V \mid f(x)=f(1-x) \text{ for } x\in [0,1]\}\]
be a subset of $V$. Determine whether the subset $W$ is a subspace of the vector space $V$.
Proof. […]