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

First, the zero element of $C(\mathbb{R})$ is the zero function $\mathbf{0}$ defined by $\mathbf{0}(x) = 0$. This element lies in $W$, as $\mathbf{0}(x) = 0 + 0 \cos(x) + 0 \cos(2x)$.

Now suppose $f_1(x), f_2(x) \in W$, say $ f_1(x) = a_1 + b_1 \cos(x) + c_1 \cos(2x)$ and $f_2(x) = a_2 + b_2 \cos(x) + c_2 \cos(2x)$. Then
\[f_1(x) + f_2(x) = (a_1 + a_2) + (b_1 + b_2) \cos(x) + ( c_1 + c_2) \cos(2x)\]
and so $f_1(x) + f_2(x) \in W$. Finally, for any scalar $d \in \mathbb{R}$, we have
\[d f_1(x) = (a_1 d) + (b_1 d) \cos(x) + (c_1 d) \cos(2x),\]
and so $d f_1(x) \in W$ as well.

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 $B=\{\sin^2(x), \cos^2(x)\}$ […]

Cosine and Sine Functions are Linearly Independent
Let $C[-\pi, \pi]$ be the vector space of all continuous functions defined on the interval $[-\pi, \pi]$.
Show that the subset $\{\cos(x), \sin(x)\}$ in $C[-\pi, \pi]$ is linearly independent.
Proof.
Note that the zero vector in the vector space $C[-\pi, \pi]$ is […]

Determine the Values of $a$ so that $W_a$ is a Subspace
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?
Solution.
The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by […]

Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent?
Let $C[-2\pi, 2\pi]$ be the vector space of all continuous functions defined on the interval $[-2\pi, 2\pi]$.
Consider the functions \[f(x)=\sin^2(x) \text{ and } g(x)=\cos^2(x)\]
in $C[-2\pi, 2\pi]$.
Prove or disprove that the functions $f(x)$ and $g(x)$ are linearly […]

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 […]

Determine Trigonometric Functions with Given Conditions
(a) Find a function
\[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)\]
such that $g(0) = g(\pi/2) = g(\pi) = 0$, where $a, b, c$ are constants.
(b) Find real numbers $a, b, c$ such that the function
\[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 […]

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 […]