Determine Whether Given Subsets in $\R^4$ are Subspaces or Not

Ohio State University exam problems and solutions in mathematics

Problem 480

(a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}$ satisfying
\[2x+4y+3z+7w+1=0.\] Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(b) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}$ satisfying
\[2x+4y+3z+7w=0.\] Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(These two problems look similar but note that the equations are different.)

(The Ohio State University, Linear Algebra Final Exam Problem)
 
LoadingAdd to solve later

Sponsored Links


Solution.

(a) $2x+4y+3z+7w+1=0$

We claim that $S$ is not a subspace of $\R^4$.
If $S$ is a subspace of $\R^4$, then the zero vector $\mathbf{0}=\begin{bmatrix}
0 \\
0 \\
0 \\
0
\end{bmatrix}$ in $\R^4$ must lie in $S$.

However, the zero vector $\mathbf{0}$ does not satisfy the equation
\[2x+4y+3z+7w+1=0.\]

So $\mathbf{0} \not \in S$, and we conclude that $S$ is not subspace of $\R^4$.

(b) $2x+4y+3z+7w=0$

In a set theoretical notation, we have
\[S=\left\{\, \begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}\in \R^4 \quad \middle| \quad 2x+4y+3z+7w=0 \,\right\}.\]

Let $A$ be the $1\times 4$ matrix defined by
\[A=\begin{bmatrix}
2 & 4 & 3 & 7
\end{bmatrix}.\] Then the equation $2x+4y+3z+7w=0$ can be written as
\[A\begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}=0.\]

So we have
\begin{align*}
S&=\left\{\, \begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}\in \R^4 \quad \middle| \quad A\begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}=0 \,\right\}\\
&=\calN(A),
\end{align*}
the null space of $A$.

Recall that the null space of a matrix is always a subspace.
Hence the subset $S$ is a subspace of $\R^4$ as it is the null space of the matrix $A$.

Final Exam Problems and Solution. (Linear Algebra Math 2568 at the Ohio State University)

This problem is one of the final exam problems of Linear Algebra course at the Ohio State University (Math 2568).

The other problems can be found from the links below.

  1. Find All the Eigenvalues of 4 by 4 Matrix
  2. Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue
  3. Diagonalize a 2 by 2 Matrix if Diagonalizable
  4. Find an Orthonormal Basis of the Range of a Linear Transformation
  5. The Product of Two Nonsingular Matrices is Nonsingular
  6. Determine Whether Given Subsets in ℝ4 R 4 are Subspaces or Not (This page)
  7. Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials
  8. Find Values of $a , b , c$ such that the Given Matrix is Diagonalizable
  9. Idempotent Matrix and its Eigenvalues
  10. Diagonalize the 3 by 3 Matrix Whose Entries are All One
  11. Given the Characteristic Polynomial, Find the Rank of the Matrix
  12. Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$
  13. Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$

LoadingAdd to solve later

Sponsored Links

More from my site

  • Hyperplane Through Origin is Subspace of 4-Dimensional Vector SpaceHyperplane Through Origin is Subspace of 4-Dimensional Vector Space Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying \[2x+3y+5z+7w=0.\] Then prove that the set $S$ is a subspace of $\R^4$. (Linear Algebra Exam Problem, The Ohio State […]
  • Subspace of Skew-Symmetric Matrices and Its DimensionSubspace of Skew-Symmetric Matrices and Its Dimension Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.) (a) Prove that the subset $W$ is a subspace of $V$. (b) Find the […]
  • Vector Space of 2 by 2 Traceless MatricesVector Space of 2 by 2 Traceless Matrices Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers. Let \[W=\left\{\, A\in V \quad \middle | \quad A=\begin{bmatrix} a & b\\ c& -a \end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\] (a) Show that $W$ is a subspace of […]
  • Maximize the Dimension of the Null Space of $A-aI$Maximize the Dimension of the Null Space of $A-aI$ Let \[ A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.\] Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix. Your score of this problem is equal to that […]
  • Orthonormal Basis of Null Space and Row SpaceOrthonormal Basis of Null Space and Row Space Let $A=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}$. (a) Find an orthonormal basis of the null space of $A$. (b) Find the rank of $A$. (c) Find an orthonormal basis of the row space of $A$. (The Ohio State University, Linear Algebra Exam […]
  • Given All Eigenvalues and Eigenspaces, Compute a Matrix ProductGiven All Eigenvalues and Eigenspaces, Compute a Matrix Product Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces \[E_2=\Span\left \{\quad \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix} 1 \\ 2 \\ 1 \\ 1 […]
  • True or False Problems of Vector Spaces and Linear TransformationsTrue or False Problems of Vector Spaces and Linear Transformations These are True or False problems. For each of the following statements, determine if it contains a wrong information or not. Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$. The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because […]
  • Find an Orthonormal Basis of the Given Two Dimensional Vector SpaceFind an Orthonormal Basis of the Given Two Dimensional Vector Space Let $W$ be a subspace of $\R^4$ with a basis \[\left\{\, \begin{bmatrix} 1 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix} \,\right\}.\] Find an orthonormal basis of $W$. (The Ohio State […]

You may also like...

3 Responses

  1. 06/28/2017

    […] Determine Wether Given Subsets in ℝ4 R 4 are Subspaces or Not […]

  2. 08/17/2017

    […] Determine Wether Given Subsets in ℝ4 R 4 are Subspaces or Not […]

  3. 11/18/2017

    […] Determine Whether Given Subsets in ℝ4 R 4 are Subspaces or Not […]

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
Ohio State University exam problems and solutions in mathematics
The Product of Two Nonsingular Matrices is Nonsingular

Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix....

Close