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)
 
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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

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