Vector Spaces and Subspaces

A nonempty set $S$ is called a vector space if it satisfies the following conditions: (i) For any $\mathbf{x}, \mathbf{y}$ in $S, \mathbf{x}+\mathbf{y}$ is defined and is in $S$. Furthermore, $$ \begin{aligned} \mathbf{x}+\mathbf{y} &=\mathbf{y}+\mathbf{x}, \quad \text { (commutativity) } \ \mathbf{x}+(\mathbf{y}+\mathbf{z}) &=(\mathbf{x}+\mathbf{y})+\mathbf{z} . \quad \text { (associativity) } \end{aligned} $$ (ii) There exists an element in $S$, denoted by $\mathbf{0}$, such that $\mathbf{x}+\mathbf{0}=\mathbf{x}$ for all $\mathbf{x}$. (iii) For any $\mathbf{x}$ in $S$, there exists an element $\mathbf{y}$ in $S$ such that $\mathbf{x}+\mathbf{y}=\mathbf{0}$. (iv) For any $\mathbf{x}$ in $S$ and any real number $c, c \mathbf{x}$ is defined and is in $S ;$ moreover, $1 \mathbf{x}=\mathbf{x}$ for any $\mathbf{x}$ (v) For any $\mathbf{x}{\mathbf{1}}, \mathbf{x}{\mathbf{2}}$ in $S$ and real numbers $c_{1}, c_{2}, c_{1}\left(\mathbf{x}{1}+\mathbf{x}{\mathbf{2}}\right)=c_{1} \mathbf{x}{\mathbf{1}}+c{1} \mathbf{x}{\mathbf{2}},\left(c{1}+\right.$ $\left.c_{2}\right) \mathbf{x}{\mathbf{1}}=c{1} \mathbf{x}{\mathbf{1}}+c{2} \mathbf{x}{\mathbf{1}}$ and $c{1}\left(c_{2} \mathbf{x}{\mathbf{1}}\right)=\left(c{1} c_{2}\right) \mathbf{x}_{\mathbf{1}}$

Elements in $S$ are called vectors. If $\mathbf{x}, \mathbf{y}$ are vectors, then the operation of taking their sum $\mathbf{x}+\mathbf{y}$ is referred to as vector addition. The vector in (ii) is called the $z$ ero vector. The operation in (iv) is called scalar multiplication. A vector space may be defined with reference to any field. We have taken the field to be the field of real numbers as this will be sufficient for our purpose.

The set of column vectors of order $n$ (or $n \times 1$ matrices) is a vector space. So is the set of row vectors of order $n$. These two vector spaces are the ones we consider most of the time. Let $R^{n}$ denote the set $R \times R \times \cdots \times R$, taken $n$ times, where $R$ is the set of real numbers. We will write elements of $R^{n}$ either as column vectors or as row vectors depending upon whichever is convenient in a given situation. If $S, T$ are vector spaces and $S \subset T$, then $S$ is called a subspace of $T$. Let us describe all possible subspaces of $R^{3}$. Clearly, $R^{3}$ is a vector space, and so is the space consisting of only the zero vector, i.e., the vector of all zeros. Let $c_{1}, c_{2}, c_{3}$ be real numbers. The set of all vectors $\mathbf{x} \in R^{3}$ that satisfy $$ c_{1} x_{1}+c_{2} x_{2}+c_{3} x_{3}=0 $$ is a subspace of $R^{3}$ (Here $x_{1}, x_{2}, x_{3}$ are the coordinates of $\left.\mathbf{x}\right)$. Geometrically, this set represents a plane passing through the origin. Intersection of two distinct planes through the origin is a straight line through the origin and is also a subspace. These are the only possible subspaces of $R^{3}$


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