# Chapter 4 - Vector Spaces - 4.5 Basis and Dimension - 4.5 Exercises - Page 188: 76

(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 $R^{3}$.

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