# 12 Examples of Subsets that Are Not Subspaces of Vector Spaces

## Problem 338

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 the vector space $\R^3$.

**(2)** \[S_2=\left \{\, \begin{bmatrix}

x_1 \\

x_2 \\

x_3

\end{bmatrix} \in \R^3 \quad \middle | \quad x_1-4x_2+5x_3=2 \,\right \}\]
in the vector space $\R^3$.

**(3)** \[S_3=\left \{\, \begin{bmatrix}

x \\

y

\end{bmatrix}\in \R^2 \quad \middle | \quad y=x^2 \quad \,\right \}\]
in the vector space $\R^2$.

**(4)** Let $P_4$ be the vector space of all polynomials of degree $4$ or less with real coefficients.

\[S_4=\{ f(x)\in P_4 \mid f(1) \text{ is an integer}\}\]
in the vector space $P_4$.

**(5)** \[S_5=\{ f(x)\in P_4 \mid f(1) \text{ is a rational number}\}\]
in the vector space $P_4$.

**(6)** Let $M_{2 \times 2}$ be the vector space of all $2\times 2$ real matrices.

\[S_6=\{ A\in M_{2\times 2} \mid \det(A) \neq 0\} \]
in the vector space $M_{2\times 2}$.

**(7)** \[S_7=\{ A\in M_{2\times 2} \mid \det(A)=0\} \]
in the vector space $M_{2\times 2}$.

(*Linear Algebra Exam Problem, the Ohio State University*)

**(8)** Let $C[-1, 1]$ be the vector space of all real continuous functions defined on the interval $[a, b]$.

\[S_8=\{ f(x)\in C[-2,2] \mid f(-1)f(1)=0\} \]
in the vector space $C[-2, 2]$.

**(9)** \[S_9=\{ f(x) \in C[-1, 1] \mid f(x)\geq 0 \text{ for all } -1\leq x \leq 1\}\]
in the vector space $C[-1, 1]$.

**(10)** Let $C^2[a, b]$ be the vector space of all real-valued functions $f(x)$ defined on $[a, b]$, where $f(x), f'(x)$, and $f^{\prime\prime}(x)$ are continuous on $[a, b]$. Here $f'(x), f^{\prime\prime}(x)$ are the first and second derivative of $f(x)$.

\[S_{10}=\{ f(x) \in C^2[-1, 1] \mid f^{\prime\prime}(x)+f(x)=\sin(x) \text{ for all } -1\leq x \leq 1\}\]
in the vector space $C[-1, 1]$.

**(11)** Let $S_{11}$ be the set of real polynomials of degree exactly $k$, where $k \geq 1$ is an integer, in the vector space $P_k$.

**(12)** Let $V$ be a vector space and $W \subset V$ a vector subspace. Define the subset $S_{12}$ to be the **complement** of $W$,

\[ V \setminus W = \{ \mathbf{v} \in V \mid \mathbf{v} \not\in W \}.\]

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Contents

- Problem 338
- Solution.
- Solution (1). $S_1=\{ \mathbf{x} \in \R^3 \mid x_1\geq 0 \}$
- Solution (2). $S_2= \{ \mathbf{x}\in \R^3\mid x_1-4x_2+5x_3=2 \}$
- Solution (3). $S_3=\{\mathbf{x}\in \R^2 \mid y=x^2 \quad \}$
- Solution (4). $S_4=\{ f(x)\in P_4 \mid f(1) \text{ is an integer}\}$
- Solution (5). $S_5=\{ f(x)\in P_4 \mid f(1) \text{ is a rational number}\}$
- Solution (6). $S_6=\{ A\in M_{2\times 2} \mid \det(A) \neq 0\}$
- Solution (7). $S_7=\{ A\in M_{2\times 2} \mid \det(A)=0\}$
- Solution (8). $S_8=\{ f(x)\in C[-2,2] \mid f(-1)f(1)=0\}$
- Solution (9). $S_9=\{ f(x) \in C[-1, 1] \mid f(x)\geq 0 \text{ for all } -1\leq x \leq 1\}$
- Solution (10). $S_{10}=\{ f(x) \in C^2[-1, 1] \mid f^{\prime\prime}(x)+f(x)=\sin(x) \text{ for all } -1\leq x \leq 1\}$
- Solution (11). Let $S_{11}$ be the set of real polynomials of degree exactly $k$.
- Solution (12). The complement

- Linear Algebra Midterm Exam 2 Problems and Solutions

## Solution.

Recall the following subspace criteria.

A subset $W$ of a vector space $V$ over the scalar field $K$ is a subspace of $V$ if and only if the following three criteria are met.

- The subset $W$ contains the zero vector of $V$.
- If $u, v\in W$, then $u+v\in W$.
- If $u\in W$ and $a\in K$, then $au\in W$.

Thus, to prove a subset $W$ is not a subspace, we just need to find a counterexample of any of the three criteria.

### Solution (1). $S_1=\{ \mathbf{x} \in \R^3 \mid x_1\geq 0 \}$

The subset $S_1$ does not satisfy condition 3. For example, consider the vector

\[\mathbf{x}=\begin{bmatrix}

1 \\

0 \\

0

\end{bmatrix}.\]
Then since $x_1=1\geq 0$, the vector $\mathbf{x}\in S_1$. Then consider the scalar product of $\mathbf{x}$ and the scalar $-1$. Then we have

\[(-1)\cdot\mathbf{x}=\begin{bmatrix}

-1 \\

0 \\

0

\end{bmatrix},\]
and the first entry is $-1$, hence $-\mathbf{x}$ is not in $S_1$. Thus $S_1$ does not satisfy condition 3 and it is not a subspace of $\R^3$.

(You can check that conditions 1, 2 are met.)

### Solution (2). $S_2= \{ \mathbf{x}\in \R^3\mid x_1-4x_2+5x_3=2 \}$

The zero vector of the vector space $\R^3$ is

\[\mathbf{0}=\begin{bmatrix}

0 \\

0 \\

0

\end{bmatrix}.\]
Since the zero vector $\mathbf{0}$ does not satisfy the defining relation $x_1-4x_2+5x_3=2$, it is not in $S_2$. Hence condition 1 is not met, hence $S_2$ is not a subspace of $\R^3$.

(You can check that conditions 2, 3 are not met as well.)

### Solution (3). $S_3=\{\mathbf{x}\in \R^2 \mid y=x^2 \quad \}$

Consider vectors

\[\begin{bmatrix}

1 \\

1

\end{bmatrix} \text{ and } \begin{bmatrix}

-1 \\

1

\end{bmatrix}.\]
These are vectors in $S_3$ since both vectors satisfy the defining relation $y=x^2$.

However, their sum

\[\begin{bmatrix}

1 \\

1

\end{bmatrix} + \begin{bmatrix}

-1 \\

1

\end{bmatrix}

=

\begin{bmatrix}

0 \\

1

\end{bmatrix}\]
is not in $S_3$ since $1\neq 0^2$.

Hence condition 2 is not met, and thus $S_3$ is not a subspace of $\R^2$.

(You can check that condition 1 is fulfilled yet condition 3 is not met.)

### Solution (4). $S_4=\{ f(x)\in P_4 \mid f(1) \text{ is an integer}\}$

Consider the polynomial $f(x)=x$. Since the degree of $f(x)$ is $1$ and $f(1)=1$ is an integer, it is in $S_4$. Consider the scalar product of $f(x)$ and the scalar $1/2\in \R$.

Then we evaluate the scalar product at $x=1$ and we have

\begin{align*}

\frac{1}{2}f(1)=\frac{1}{2},

\end{align*}

which is not an integer.

Thus $(1/2)f(x)$ is not in $S_4$, hence condition 3 is not met. Thus $S_4$ is not a subspace of $P_4$.

(You can check that conditions 1, 2 are met.)

### Solution (5). $S_5=\{ f(x)\in P_4 \mid f(1) \text{ is a rational number}\}$

Let $f(x)=x$. Then $f(x)$ is a degree $1$ polynomial and $f(1)=1$ is a rational number.

However, the scalar product $\sqrt{2} f(x)$ of $f(x)$ and the scalar $\sqrt{2} \in \R$ is not in $S_5$ since

\[\sqrt{2}f(1)=\sqrt{2},\]
which is not a rational number. Hence condition 3 is not met and $S_5$ is not a subspace of $P_4$.

(You can check that conditions 1, 2 are met.)

### Solution (6). $S_6=\{ A\in M_{2\times 2} \mid \det(A) \neq 0\}$

The zero vector of the vector space $M_{2 \times 2}$ is the $2\times 2$ zero matrix $O$.

Since the determinant of the zero matrix $O$ is $0$, it is not in $S_6$. Thus, condition 1 is not met and $S_6$ is not a subspace of $M_{2 \times 2}$.

(You can check that conditions 2, 3 are not met as well.)

### Solution (7). $S_7=\{ A\in M_{2\times 2} \mid \det(A)=0\}$

Consider the matrices

\[A=\begin{bmatrix}

1 & 0\\

0& 0

\end{bmatrix} \text{ and } B=\begin{bmatrix}

0 & 0\\

0& 1

\end{bmatrix}.\]
The determinants of $A$ and $B$ are both $0$, hence they belong to $S_7$.

However, their sum

\[A+B=\begin{bmatrix}

1 & 0\\

0& 1

\end{bmatrix}\]
has the determinant $1$, hence the sum $A+B$ is not in $S_7$.

So condition 2 is not met and $S_7$ is not a subspace of $M_{2 \times 2}$.

(You can check that conditions 1, 3 are met.)

### Solution (8). $S_8=\{ f(x)\in C[-2,2] \mid f(-1)f(1)=0\}$

Consider the continuous functions

\[f(x)=x-1 \text{ and } g(x)=x+1.\]
(These are polynomials, hence they are continuous.)

We have

\begin{align*}

&f(-1)f(1)=(-2)\cdot(0)=0 \text{ and }\\

&g(-1)g(1)=(0)\cdot 2=0.

\end{align*}

So these functions are in $S_8$.

However, their sum $h(x):=f(x)+g(x)$ does not belong to $S_8$ since we have

\begin{align*}

h(-1)h(1)&=\big(f(-1)+g(-1)\big) \big(f(1)+g(1) \big)\\

&=(-2+0)(0+2)=-4\neq 0.

\end{align*}

Therefore, condition 2 is not met and $S_8$ is not a subspace of $C[-1, 1]$.

(You can check that conditions 1, 3 are met.)

### Solution (9). $S_9=\{ f(x) \in C[-1, 1] \mid f(x)\geq 0 \text{ for all } -1\leq x \leq 1\}$

Let $f(x)=x^2$, an open-up parabola.

Then $f(x)$ is continuous and non-negative for $-1 \leq x \leq 1$. Hence $f(x)=x^2$ is in $S_9$.

However, the scalar product $(-1)f(x)$ of $f(x)$ and the scalar $-1$ is not in $S_9$ since, say,

\[(-1)f(1)=-1\]
is negative.

So condition 3 is not met and $S_9$ is not a subspace of $C[-1, 1]$.

(You can check that conditions 1, 2 are met.)

### Solution (10). $S_{10}=\{ f(x) \in C^2[-1, 1] \mid f^{\prime\prime}(x)+f(x)=\sin(x) \text{ for all } -1\leq x \leq 1\}$

The zero vector of the vector space $C^2[-1, 1]$ is the zero function $\theta(x)=0$.

The second derivative of the zero function is still the zero function.

Thus,

\[\theta^{\prime\prime}(x)+\theta(x)=0\]
and since $\sin(x)$ is not the zero function, $\theta(x)$ is not in $S_{10}$.

Hence $S_{10}$ is not a subspace of $C^2[-1, 1]$.

(You can check that conditions 2, 3 are not met as well.

For example, consider the function $f(x)=-\frac{1}{2}x\cos(x)\in S_{10}$.)

### Solution (11). Let $S_{11}$ be the set of real polynomials of degree exactly $k$.

The set $S_{11}$ is not a vector subspace of $\mathbf{P}_k$. One reason is that the zero function $\mathbf{0}$ has degree $0$, and so does not lie in $S_{11}$. The set $S_{11}$ is also not closed under addition. Consider the two polynomials $f(x) = x^k + 1$ and $g(x) = – x^k + 1$. Both of these polynomials lie in $S_{11}$, however $f(x) + g(x) = 2$ has degree $0$ and so does not lie in $S_{11}$.

### Solution (12). The complement

The complement $S_{12}= V \setminus W$ is not a vector subspace. Specifically, if $\mathbf{0} \in V$ is the zero vector, then we know $\mathbf{0} \in W$ because $W$ is a subspace. But then $\mathbf{0} \not\in V \setminus W$, and so $V \setminus W$ cannot be a vector subspace.

## Linear Algebra Midterm Exam 2 Problems and Solutions

- True of False Problems and Solutions: True or False problems of vector spaces and linear transformations
- Problem 1 and its solution (current problem): See (7) in the post “10 examples of subsets that are not subspaces of vector spaces”
- Problem 2 and its solution: Determine whether trigonometry functions $\sin^2(x), \cos^2(x), 1$ are linearly independent or dependent
- Problem 3 and its solution: Orthonormal basis of null space and row space
- Problem 4 and its solution: Basis of span in vector space of polynomials of degree 2 or less
- Problem 5 and its solution: Determine value of linear transformation from $R^3$ to $R^2$
- Problem 6 and its solution: Rank and nullity of linear transformation from $R^3$ to $R^2$
- Problem 7 and its solution: Find matrix representation of linear transformation from $R^2$ to $R^2$
- Problem 8 and its solution: Hyperplane through origin is subspace of 4-dimensional vector space

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