# Determine if the set of vectors $\{[-1, 3, 1], [2, 1, 4]\}$ is a basis for the subspace of $\mathbb{R}^3$ that the vectors span.

I recognize that this question is essentially identical to Determine whether the set of vectors is a basis for the subspace of $$\mathbb{R}^n$$that the vectors span posted by user K Split X; their question comes from a modified version of Question 29 on Chapter 1.6 on John B. Fraleigh and Raymond A. Beauregard's Linear Algebra textbook. Upon reviewing their question, I have made a solution attempt at the problem but have failed to reach the answer provided by the textbook; I would like to know why.

I determined that the above set of vectors should form a basis for the subspace of $$\mathbb{R}^3$$ that the vectors span; the textbook states otherwise.

My reasoning is as follows:

According to Beauregard, given an $$m$$ $$\times$$ $$n$$ matrix $$A$$, the following are equivalent:

1. Each consistent system $$A\vec{x}=\vec{b}$$ has a unique solution
2. The reduced row-echelon form of $$A$$ consists of the $$n$$ $$\times$$ $$n$$ identity matrix followed by $$m-n$$ rows of zeros
3. The column vectors of $$A$$ form a basis for the column space of $$A$$

Thus, I concluded that, given $$m$$ row vectors of dimensions $$1$$ $$\times$$ $$n$$, the set of vectors form a basis for the subspace of $$\mathbb{R}^n$$that the vectors span if and only if the matrix A formed by the transposes of the set of vectors can be row-reduced into the $$m$$ $$\times$$ $$m$$ identity matrix followed by $$n-m$$ rows of zeros.

Because matrix A = $$\begin{bmatrix}-1&2\\3&1\\1&4\end{bmatrix}$$ can be reduced into $$\begin{bmatrix}1&0\\0&1\\0&0\end{bmatrix}$$, the above statement is satisfied and the column vectors of $$A$$ form a basis for the column space of $$A$$. Since the column vectors of $$A$$ form a basis for the column space of $$A$$, the set of vectors $$\{[-1, 3, 1], [2, 1, 4]\}$$ is a basis for the subspace of $$\mathbb{R}^3$$ that the vectors span.

The textbook states that the set of vectors is not a basis. May someone enlighten me as to where I have made a mistake? Thank you so much!

## 1 Answer

There's just an error in the textbook, your reasoning is solid. Also the Author's names are John B. Fraleigh and Raymond A. Beauregard, which you seem to have combined into a single person Francis Beauregard.

• Thank you. Thanks for the correction.
– user971909
Commented Mar 6, 2022 at 18:55