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 […]
  • 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 […]
  • 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 […]
  • 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 […]
  • 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 \[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\] […]

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 *

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