Question

Determine which of the following sets of vectors are bases of $\mathbb{R}^2$ : (a) $\left(\begin{array}{r}1 \\ -3\end{array}\right),\left(\begin{array}{r}-2 \\ 5\end{array}\right)$ (b) $\left(\begin{array}{r}1 \\ -1\end{array}\right),\left(\begin{array}{r}-1 \\ 1\end{array}\right)$; (c) $\left(\begin{array}{l}1 \\ 2\end{array}\right),\left(\begin{array}{l}2 \\ 1\end{array}\right)$; (d) $\left(\begin{array}{l}3 \\ 5\end{array}\right),\left(\begin{array}{l}0 \\ 0\end{array}\right)$; (e) $\left(\begin{array}{l}2 \\ 0\end{array}\right),\left(\begin{array}{r}-1 \\ 2\end{array}\right),\left(\begin{array}{r}0 \\ -1\end{array}\right)$.

    Determine which of the following sets of vectors are bases of $\mathbb{R}^2$ :
(a) $\left(\begin{array}{r}1 \\ -3\end{array}\right),\left(\begin{array}{r}-2 \\ 5\end{array}\right)$
(b) $\left(\begin{array}{r}1 \\ -1\end{array}\right),\left(\begin{array}{r}-1 \\ 1\end{array}\right)$;
(c) $\left(\begin{array}{l}1 \\ 2\end{array}\right),\left(\begin{array}{l}2 \\ 1\end{array}\right)$;
(d) $\left(\begin{array}{l}3 \\ 5\end{array}\right),\left(\begin{array}{l}0 \\ 0\end{array}\right)$;
(e) $\left(\begin{array}{l}2 \\ 0\end{array}\right),\left(\begin{array}{r}-1 \\ 2\end{array}\right),\left(\begin{array}{r}0 \\ -1\end{array}\right)$.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 2, Problem 1

Instant Answer

Step 1

A set of vectors in $\mathbb{R}^2$ forms a basis if and only if they are linearly independent and span $\mathbb{R}^2$. Since $\mathbb{R}^2$ is a 2-dimensional vector space, any set of two vectors that are linearly independent will automatically span Show more…

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Determine which of the following sets of vectors are bases of $\mathbb{R}^2$ : (a) $\left(\begin{array}{r}1 \\ -3\end{array}\right),\left(\begin{array}{r}-2 \\ 5\end{array}\right)$ (b) $\left(\begin{array}{r}1 \\ -1\end{array}\right),\left(\begin{array}{r}-1 \\ 1\end{array}\right)$; (c) $\left(\begin{array}{l}1 \\ 2\end{array}\right),\left(\begin{array}{l}2 \\ 1\end{array}\right)$; (d) $\left(\begin{array}{l}3 \\ 5\end{array}\right),\left(\begin{array}{l}0 \\ 0\end{array}\right)$; (e) $\left(\begin{array}{l}2 \\ 0\end{array}\right),\left(\begin{array}{r}-1 \\ 2\end{array}\right),\left(\begin{array}{r}0 \\ -1\end{array}\right)$.

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