Definition and Theorems of Chapter 2

Definition. A vector space (or linear space) consists of the following:

1. a field $F$ of scalars;
2. a set $V$ of objects, called vectors;
3. a rule (or operation), called vector addition, which associates with each pair of vectors ${\alpha},{\beta}$ in $V$ a vector $\alpha+\beta$ in $V$, called the sum of ${\alpha}$ and ${\beta}$, in such a way that
   ( a ) addition is commutative, $\alpha+\beta=\beta+\alpha$;
   ( b ) addition is associative, ${\alpha}+(\beta +\gamma)=(\alpha+\beta)+\gamma$;
   ( c ) there is a unique vector 0 in $V$, called the zero vector, such that $\alpha+0=\alpha$ for all ${\alpha}$ in $V$;
   ( d ) for each vector ${\alpha}$ in $V$ there is a unique vector $-{\alpha}$ in $V$ such that ${\alpha}+(-{\alpha})=0$;
4. a rule (or operation), called scalar multiplication, which associates with each scalar $c$ in $F$ and vector $\alpha$ in $V$ a vector $c{\alpha}$ in $V$, called the product of $c$ and $\alpha$, in such a way that
   ( a ) $1\alpha=\alpha$ for every ${\alpha}$ in $V$;
   ( b ) $(c_1c_2)\alpha=c_1(c_2\alpha)$;
   ( c ) $c(\alpha+\beta)=c\alpha+c\beta$;
   ( d ) $(c_1+c_2)\alpha=c_1\alpha+c_2\alpha$.

Definition. A vector $\beta$ in $V$ is said to be a linear combination of the vectors ${\alpha}_1,\dots,{\alpha}_n$ in $V$ provided there exist scalars $c_1,\dots,c_n$ in $F$ such that
$\beta=c_1{\alpha}_1+\cdots+c_n{\alpha}_n=\sum_{i=1}^nc_i{\alpha}_i$

Definition. Let $V$ be a vector space over the field $F$. A subspace of $V$ is a subset $W$ of $V$ which is itself a vector space over $F$ with the operation of vector addition and scalar multiplication on $V$.

Theorem 1. A non-empty subset $W$ of $V$ is a subspace of $V$ if and only if for each pair of vectors ${\alpha},{\beta}$ in $W$ and each scalar $c$ in $F$ the vector $c\alpha+\beta$ is in $W$.

Lemma. If $A$ is an $m\times n$ matrix over $F$ and $B,C$ are $n\times p$ matrices over $F$ then
$A(dB+C)=d(AB)+AC,\quad \forall d\in F$

Theorem 2. Let $V$ be a vector space over the field $F$. Then intersection of any collection of subspaces of $V$ is a subspace of $V$.

Definition. Let $S$ be a set of vectors in a vector space $V$. The subspace spanned by $S$ is defined to be the intersection $W$ of all subspaces of $V$ which contains $S$. When $S$ is a finite set of vectors, $S=\{\alpha_1,\alpha_2,\dots,\alpha_n\}$, we shall simply call $W$ the subspace spanned by the vectors $\alpha_1,\alpha_2,\dots,\alpha_n$.

Theorem 3. The subspace spanned by a non-empty subset $S$ of a vector space $V$ is the set of all linear combinations of vectors in $S$.

Definition. If $S_1,S_2,\dots,S_k$ are subsets of a vector space $V$, the set of all sums ${\alpha}_1+{\alpha_2}+\dots+{\alpha}_k$ of vectors ${\alpha}_i$ in $S_i$ is called the sum of the subsets $S_1,S_2,\dots,S_k$ and is denoted by $S_1+S_2+\dots+S_k$ or by $\sum_{i=1}^kS_i$.

Definition. Let $V$ be a vector space over $F$. A subset $S$ of $V$ is said to be linearly dependent (or simply, dependent) if there exist distinct vectors ${\alpha}_1,{\alpha}_2,\dots,\alpha_n$ in $S$ and scalars $c_1,c_2,\dots,c_n$ in $F$, not all of which are 0, such that
$c_1{\alpha}_1+\cdots+c_n{\alpha}_n=0$
A set which is not linearly depenent is called linearly independent. If the set $S$ contains only finitely many vectors ${\alpha}_1,{\alpha}_2,\dots,\alpha_n$, we sometimes say that ${\alpha}_1,{\alpha}_2,\dots,\alpha_n$ are dependent (or independent) instead of saying $S$ is dependent (or independent).

Definition. Let $V$ be a vector space. A basis for $V$ is a linealy independent set of vectors in $V$ which spans the space $V$. The space $V$ is finite-dimensional if it has a finite basis.

Theorem 4. Let $V$ be a vector space which is spanned by a finite set of vectors ${\beta}_1,{\beta}_2,\dots,{\beta}_m$. Then any independent set of vectors in $V$ is finite and contains no more than $m$ elements.
Corollary 1. If $V$ is a finite-dimensional vector space, then any two bases of $V$ have the same (finite) number of elements.
Corollary 2. Let $V$ be a finite-dimensional vector space and let $n=\dim V$. Then
( a ) any sunset of $V$ which contains more than $n$ vectors is linearly dependent;
( b ) no subset of $V$ which contains fewer than $n$ vectors can span $V$.

Theorem 5. If $W$ is a subspace of a finite-dimensional vector space $V$, every linearly independent subset of $W$ is finite and is part of a (finite) basis for $W$.
Corollary 1. If $W$ is a proper subspace of a finite-dimensional vector space $V$, then $W$ is finite-dimensional and $\dim W<\dim V$.
Corollary 2. In a finite-dimensional vector space $V$ every non-empty linearly independent set of vectors is part of a basis.
Corollary 3. Let $A$ be an $n\times n$ matrix over a field $F$, and suppose the row vectors of $A$ form a linearly independent set of vectors in $F^n$. Then $A$ is invertible.

Theorem 6. If $W_1$ and $W_2$ are finite-dimensional subspaces of a vector space $V$, then $W_1+W_2$ is finite-dimensional and
$\dim W_1+\dim W_2=\dim(W_1\cap W_2)+\dim(W_1+W_2)$

Definition. If $V$ is a finite-dimensional vector space, an ordered basis for $V$ is a finite sequence of vectors which is linearly independent and spans $V$.

Theorem 7. Let $V$ be an $n$-dimensional vector space over the field $F$, and let $\mathfrak B$ and $\mathfrak B'$ be two ordered bases of $V$. Then there is a unique, necessarily invertible, $n\times n$ matrix $P$ with entries in $F$ such that
$[{\alpha}]_{\mathfrak B}=P[{\alpha}]_{\mathfrak B'}\qquad [{\alpha}]_{\mathfrak B'}=P^{-1}[{\alpha}]_{\mathfrak B}$
for every vector $\alpha$ in $V$. Then columns of $P$ are given by
$P_j=[{\alpha}_j']_{\mathfrak B},\qquad j=1,\dots,n$

Theorem 8. Suppose $P$ is an $n\times n$ invertilbe matrix over $F$. Let $V$ be an $n$-dimensional vector space over $F$, and let $\mathfrak B$ be an ordered basis of $V$. Then there is a unique basis $\mathfrak B'$ of $V$ such that
$[{\alpha}]_{\mathfrak B}=P[{\alpha}]_{\mathfrak B'}\qquad [{\alpha}]_{\mathfrak B'}=P^{-1}[{\alpha}]_{\mathfrak B}$
for every vector $\alpha$ in $V$.

Theorem 9. Row-equivalent matrices have the same row space.

Theorem 10. Let $R$ be a non-zero row-reduced echelon matrix. Then the non-zero row vectors of $R$ form a basis for the row space of $R$.

Theorem 11. Let $m$ and $n$ be positive integers and let $F$ be a field. Suppose $W$ is a subspace of $F^n$ and $\dim W\leq m$. Then there is precisely one $m\times n$ row-reduced echelon matrix over $F$ which has $W$ as its row space.
Corollary. Each $m\times n$ matrix $A$ is row-equivalent to one and oonly one row-reduced echelon matrix.
Corollary. Let $A$ and $B$ be $m\times n$ matrices over the field $F$. Then $A$ and $B$ are row-equivalent if and only if they have the same row space.