Hyperplane Through Origin is Subspace of 4-Dimensional Vector Space

Ohio State University exam problems and solutions in mathematics

Problem 371

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 University)
 
LoadingAdd to solve later

Sponsored Links


Proof.

First, in set theoretical notation, the definition of $S$ can be written as
\[S=\left\{\, \begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}\in \R^4 \quad \middle| \quad 2x+3y+5z+7w=0 \,\right\}.\]

Let $A=\begin{bmatrix}
2 & 3 & 5 & 7
\end{bmatrix}$ be the $1 \times 4$ matrix. Then the defining equation $2x+3y+5z+7w=0$ can be written as
\[A\mathbf{x}=0,\] where
\[\mathbf{x}=\begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}.\] It follows that the set $S$ is the null space of $A$, that is, $S=\calN(A)$.
Since every null space is a subspace, we see that $S$ is also a subspace of $\R^4$.

Linear Algebra Midterm Exam 2 Problems and Solutions


LoadingAdd to solve later

Sponsored Links

More from my site

  • Determine Whether Given Subsets in $\R^4$ are Subspaces or NotDetermine Whether Given Subsets in $\R^4$ are Subspaces or Not (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 […]
  • 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 […]
  • 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 […]
  • 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 […]
  • Quiz 5: Example and Non-Example of Subspaces in 3-Dimensional SpaceQuiz 5: Example and Non-Example of Subspaces in 3-Dimensional Space Problem 1 Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by \[W=\left\{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.\] (a) Which of the following vectors are in the subset […]
  • 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 […]
  • Hyperplane in $n$-Dimensional Space Through Origin is a SubspaceHyperplane in $n$-Dimensional Space Through Origin is a Subspace A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors \[\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\in \R^n\] satisfying the linear equation of the form \[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\] […]
  • Basis of Span in Vector Space of Polynomials of Degree 2 or LessBasis of Span in Vector Space of Polynomials of Degree 2 or Less Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients. Let \[S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}\] be the set of four vectors in $P_2$. Then find a basis of the subspace $\Span(S)$ among the vectors in $S$. (Linear […]

You may also like...

5 Responses

  1. 04/06/2017

    […] Problem 8 and its solution: Hyperplane through origin is subspace of 4-dimensional vector space […]

  2. 04/07/2017

    […] Problem 8 and its solution: Hyperplane through origin is subspace of 4-dimensional vector space […]

  3. 04/07/2017

    […] Problem 8 and its solution: Hyperplane through origin is subspace of 4-dimensional vector space […]

  4. 04/07/2017

    […] Problem 8 and its solution: Hyperplane through origin is subspace of 4-dimensional vector space […]

  5. 09/20/2017

    […] Problem 8 and its solution: Hyperplane through origin is subspace of 4-dimensional vector space […]

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
Find Matrix Representation of Linear Transformation From $\R^2$ to $\R^2$

Let $T: \R^2 \to \R^2$ be a linear transformation such that \[T\left(\, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 4 \\...

Close