Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors

Ohio State University exam problems and solutions in mathematics

Problem 606

Let $V$ be a vector space and $B$ be a basis for $V$.
Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$.
Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ with respect to the basis $B$.

After applying the elementary row operations to $A$, we obtain the following matrix in reduced row echelon form
\[\begin{bmatrix}
1 & 0 & 2 & 1 & 0 \\
0 & 1 & 3 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}.\]

(a) What is the dimension of $V$?

(b) What is the dimension of $\Span\{\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5\}$?

(The Ohio State University, Linear Algebra Midterm)
 
LoadingAdd to solve later

Solution.

(a) What is the dimension of $V$?

Suppose the dimension of $V$ is $n$.
This means that the basis $B$ consists of $n$ vectors.

Then the coordinate of $\mathbf{w}_1$ with respect to $B$ is an $n$-dimensional vector $[\mathbf{w}_1]_B \in \R^n$.
Thus, the number of rows in the matrix $A$ is $n$.

As the elementary row operations do not change the number of rows, we see from the given matrix that the $A$ has four rows.
Thus, the dimension of $V$ is $4$.

(b) What is the dimension of $\Span\{\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5\}$?

Note that the dimension of $W:=\Span\{\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5\}$ is the same as the dimension of
\[W’:=\Span\{[\mathbf{w}_1]_B, [\mathbf{w}_2]_B, [\mathbf{w}_3]_B, [\mathbf{w}_4]_B, [\mathbf{w}_5]_B\}.\]

Since the column vectors of $A$ are these coordinate vectors $[\mathbf{w}_i]_B$ and its reduced row echelon contains the leading 1’s in the first two columns, we conclude that $\{[\mathbf{w}_1]_B, [\mathbf{w}_2]_B\}$ is a basis for $W’$ by the leading 1 method.

It follows that $\{\mathbf{w}_1, \mathbf{w}_2\}$ is a basis for $W$, and its dimension is $2$.

Comment.

This is one of the midterm 2 exam problems for Linear Algebra (Math 2568) in Autumn 2017.

List of Midterm 2 Problems for Linear Algebra (Math 2568) in Autumn 2017

  1. Vector Space of 2 by 2 Traceless Matrices
  2. Find an Orthonormal Basis of the Given Two Dimensional Vector Space
  3. Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent?
  4. Find Bases for the Null Space, Range, and the Row Space of a $5\times 4$ Matrix
  5. Matrix Representation, Rank, and Nullity of a Linear Transformation $T:\R^2\to \R^3$
  6. Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors ←The current problem
  7. Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less

LoadingAdd to solve later

Sponsored Links

More from my site

  • Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or LessFind a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less. Let \[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\ p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3. \end{align*} (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 […]
  • Quiz 9. Find a Basis of the Subspace Spanned by Four MatricesQuiz 9. Find a Basis of the Subspace Spanned by Four Matrices Let $V$ be the vector space of all $2\times 2$ real matrices. Let $S=\{A_1, A_2, A_3, A_4\}$, where \[A_1=\begin{bmatrix} 1 & 2\\ -1& 3 \end{bmatrix}, A_2=\begin{bmatrix} 0 & -1\\ 1& 4 \end{bmatrix}, A_3=\begin{bmatrix} -1 & 0\\ 1& -10 \end{bmatrix}, […]
  • Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given PolynomialsFind a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less. Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where \begin{align*} p_1(x)&=-1+x+2x^2, \quad p_2(x)=x+3x^2\\ p_3(x)&=1+2x+8x^2, \quad p_4(x)=1+x+x^2. \end{align*} (a) Find […]
  • Linear Transformation and a Basis of the Vector Space $\R^3$Linear Transformation and a Basis of the Vector Space $\R^3$ Let $T$ be a linear transformation from the vector space $\R^3$ to $\R^3$. Suppose that $k=3$ is the smallest positive integer such that $T^k=\mathbf{0}$ (the zero linear transformation) and suppose that we have $\mathbf{x}\in \R^3$ such that $T^2\mathbf{x}\neq \mathbf{0}$. Show […]
  • Find Bases for the Null Space, Range, and the Row Space of a $5\times 4$ MatrixFind Bases for the Null Space, Range, and the Row Space of a $5\times 4$ Matrix Let \[A=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 &1 & 1 & 1 \\ 1 & -1 & 0 & 0 \\ 0 & 2 & 2 & 2\\ 0 & 0 & 0 & 0 \end{bmatrix}.\] (a) Find a basis for the null space $\calN(A)$. (b) Find a basis of the range $\calR(A)$. (c) Find a basis of the […]
  • Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$ Let $T:\R^3 \to \R^2$ be a linear transformation such that \[ T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 0 \\ 1 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 1 \\ 0 \end{bmatrix},\] where $\mathbf{e}_1, […]
  • 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 […]

You may also like...

3 Responses

  1. 11/08/2017

    […] Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors […]

  2. 11/08/2017

    […] Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors […]

  3. 11/08/2017

    […] Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors […]

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
Matrix Representation, Rank, and Nullity of a Linear Transformation $T:\R^2\to \R^3$

Let $T:\R^2 \to \R^3$ be a linear transformation such that \[T\left(\, \begin{bmatrix} 3 \\ 2 \end{bmatrix} \,\right) =\begin{bmatrix} 1 \\...

Close