Let $C(\mathbb{R})$ be the vector space of real-valued functions on $\mathbb{R}$.

Consider the set of functions $W = \{ f(x) = a + b \cos(x) + c \cos(2x) \mid a, b, c \in \mathbb{R} \}$.

Prove that $W$ is a vector subspace of $C(\mathbb{R})$.

Proof.

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.

This proves that $W$ is a subspace of $W$.

The post The Set $ \{ a + b \cos(x) + c \cos(2x) \mid a, b, c \in \mathbb{R} \}$ is a Subspace in $C(\R)$ first appeared on Problems in Mathematics.

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 the zero function
\[\theta(x):=0.\]

Let us consider a linear combination
\[a_1\cos(x)+a_2\sin(x)=\theta(x)=0 \tag{*}.\] If this linear combination has only the zero solution $a_1=a_2=0$, then the set $\{\cos(x), \sin(x)\}$ is linearly independent.

The equality (*) should be true for any values of $x\in [-\pi, \pi]$.
Setting $x=0$, we obtain from (*) that
\[a_1=0\] since $\cos(0)=1, \sin(0)=0$.

We also set $x=\pi/2$ and we obtain
\[a_2=0\] since $\cos(\pi/2)=0, \sin(\pi/2)=1$.

Therefore, we have $a_1=a_2=0$ and we conclude that the set $\{\cos(x), \sin(x)\}$ is linearly independent.

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