Answer
(a) A set $S_{1}$ consists of two vectors of the form $\mathbf{u}=\left(u_{1}, u_{2}, u_{3}\right) $. $S_{1}$ is not a basis
for $R^{3}$ because it contains only two vectors but any basis for $R^3$ must contain three vectors.
(b) A set $S_{2}$ consists of four vectors of the form
$\mathbf{u}=\left(u_{1}, u_{2}, u_{3}\right) $. $S_{2}$ is not a basis because it contains four vectors but any basis for $R^3$ must be three vectors.
(c) A set $S_{3}$ consists of three vectors of the form
$\mathbf{u}=\left(u_{1}, u_{2}, u_{3}\right) $. The conditions under which $S_{3}$ is a basis for $R^{3}$ is $\textbf{one} $ of the following:
1) $S_{3}$ is linearly independent.
2) $S_{3}$ spans $R^{3}$.
Work Step by Step
(a) A set $S_{1}$ consists of two vectors of the form $\mathbf{u}=\left(u_{1}, u_{2}, u_{3}\right) $. $S_{1}$ is not a basis
for $R^{3}$ because it contains only two vectors but any basis for $R^3$ must contain three vectors.
(b) A set $S_{2}$ consists of four vectors of the form
$\mathbf{u}=\left(u_{1}, u_{2}, u_{3}\right) $. $S_{2}$ is not a basis because it contains four vectors but any basis for $R^3$ must be three vectors.
(c) A set $S_{3}$ consists of three vectors of the form
$\mathbf{u}=\left(u_{1}, u_{2}, u_{3}\right) $. The conditions under which $S_{3}$ is a basis for $R^{3}$ is $\textbf{one} $ of the following:
1) $S_{3}$ is linearly independent.
2) $S_{3}$ spans $R^{3}$.
