Subspaces of the Vector Space of All Real Valued Function on the Interval

Linear algebra problems and solutions

Problem 134

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) $T=\{f(x) \in V \mid f(0)=f(1)+3\}$.

 
LoadingAdd to solve later

Sponsored Links


 

Hint.

To show that a subset $W$ of a vector space $V$ is a subspace, we need to check that

  1. the zero vector in $V$ is in $W$
  2. for any two vectors $u,v \in W$, we have $u+v \in W$
  3. for any scalar $c$ and any vector $u \in W$, we have $cu \in W$.

Solution.

(a) Is $S=\{f(x) \in V \mid f(0)=f(1)\}$ a subspace?

We show that $S$ is a subspace of the vector space $V$ by checking conditions (1)-(3) given in the hint above.
First note that the zero vector in $V$ is the zero function $\theta(x)$, that is, $\theta(x)=0$ for any $x \in [0,1]$.
Since we have $\theta(0)=0=\theta(1)$, the zero function $\theta(x)\in S$.
Condition (1) is met.

Now, take any $f(x), g(x) \in S$. By the defining relation of $S$, we have
\[f(0)=f(1), \quad g(0)=g(1).\] Consider the addition $(f+g)(x)$. We have
\[(f+g)(0)=f(0)+g(0)=f(1)+g(1)=(f+g)(1)\] and it follows that $(f+g)(x) \in S$.
Thus $S$ satisfies condition (2).

To check the condition (3), take any scalar $c \in \R$ and $f(x) \in S$.
Since $f(x)\in S$, we have $f(0)=f(1)$. The scalar multiplication $(cf)(x)$ satisfies
\[(cf)(0)=c\cdot f(0)=c\cdot f(1)= (cf)(0).\] Thus $(cf)(x) \in S$.

Therefore, the subset $S$ satisfies conditions (1)-(3). Hence $S$ is a subspace of $V$.

(b) Is $T=\{f(x) \in V \mid f(0)=f(1)+3\}$ a subspace?

We claim that $T$ is not a subspace of the vector space $V$.
For example, the subset $T$ does not satisfy condition (1).

The zero vector of $V$ is the zero function $\theta(x)$.
Then we have
\[\theta(0)=0 \neq 0+3=\theta(1)+3,\] and hence the zero vector $\theta(x) \in V$ is not in $W$.


LoadingAdd to solve later

Sponsored Links

More from my site

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Linear Algebra exam problems and solutions at University of California, Berkeley
Square Root of an Upper Triangular Matrix. How Many Square Roots Exist?

Find a square root of the matrix \[A=\begin{bmatrix} 1 & 3 & -3 \\ 0 &4 &5 \\ 0 &...

Close