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

1. a field $F$F of scalars;
2. a set $V$V of objects, called vectors;
3. a rule (or operation), called vector addition, which associates with each pair of vectors ${\alpha},{\beta}$α,β in $V$V a vector $\alpha+\beta$α+β in $V$V, called the sum of ${\alpha}$α and ${\beta}$β, in such a way that
   ( a ) addition is commutative, $\alpha+\beta=\beta+\alpha$α+β=β+α;
   ( b ) addition is associative, ${\alpha}+(\beta +\gamma)=(\alpha+\beta)+\gamma$α+(β+γ)=(α+β)+γ;
   ( c ) there is a unique vector 0 in $V$V, called the zero vector, such that $\alpha+0=\alpha$α+0=α for all ${\alpha}$α in $V$V;
   ( d ) for each vector ${\alpha}$α in $V$V there is a unique vector $-{\alpha}$−α in $V$V such that ${\alpha}+(-{\alpha})=0$α+(−α)=0;
4. a rule (or operation), called scalar multiplication, which associates with each scalar $c$c in $F$F and vector $\alpha$α in $V$V a vector $c{\alpha}$cα in $V$V, called the product of $c$c and $\alpha$α, in such a way that
   ( a ) $1\alpha=\alpha$1α=α for every ${\alpha}$α in $V$V;
   ( b ) $(c_1c_2)\alpha=c_1(c_2\alpha)$(c1​c2​)α=c1​(c2​α);
   ( c ) $c(\alpha+\beta)=c\alpha+c\beta$c(α+β)=cα+cβ;
   ( d ) $(c_1+c_2)\alpha=c_1\alpha+c_2\alpha$(c1​+c2​)α=c1​α+c2​α.

Definition. A vector $\beta$β in $V$V is said to be a linear combination of the vectors ${\alpha}_1,\dots,{\alpha}_n$α1​,…,αn​ in $V$V provided there exist scalars $c_1,\dots,c_n$c1​,…,cn​ in $F$F such that
$\beta=c_1{\alpha}_1+\cdots+c_n{\alpha}_n=\sum_{i=1}^nc_i{\alpha}_i$β=c1​α1​+⋯+cn​αn​=i=1∑n​ci​αi​

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

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

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

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

Definition. Let $S$S be a set of vectors in a vector space $V$V. The subspace spanned by $S$S is defined to be the intersection $W$W of all subspaces of $V$V which contains $S$S. When $S$S is a finite set of vectors, $S=\{\alpha_1,\alpha_2,\dots,\alpha_n\}$S={α1​,α2​,…,αn​}, we shall simply call $W$W the subspace spanned by the vectors $\alpha_1,\alpha_2,\dots,\alpha_n$α1​,α2​,…,αn​.

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

Definition. If $S_1,S_2,\dots,S_k$S1​,S2​,…,Sk​ are subsets of a vector space $V$V, the set of all sums ${\alpha}_1+{\alpha_2}+\dots+{\alpha}_k$α1​+α2​+⋯+αk​ of vectors ${\alpha}_i$αi​ in $S_i$Si​ is called the sum of the subsets $S_1,S_2,\dots,S_k$S1​,S2​,…,Sk​ and is denoted by $S_1+S_2+\dots+S_k$S1​+S2​+⋯+Sk​ or by $\sum_{i=1}^kS_i$∑i=1k​Si​.

Definition. Let $V$V be a vector space over $F$F. A subset $S$S of $V$V is said to be linearly dependent (or simply, dependent) if there exist distinct vectors ${\alpha}_1,{\alpha}_2,\dots,\alpha_n$α1​,α2​,…,αn​ in $S$S and scalars $c_1,c_2,\dots,c_n$c1​,c2​,…,cn​ in $F$F, not all of which are 0, such that
$c_1{\alpha}_1+\cdots+c_n{\alpha}_n=0$c1​α1​+⋯+cn​αn​=0
A set which is not linearly depenent is called linearly independent. If the set $S$S contains only finitely many vectors ${\alpha}_1,{\alpha}_2,\dots,\alpha_n$α1​,α2​,…,αn​, we sometimes say that ${\alpha}_1,{\alpha}_2,\dots,\alpha_n$α1​,α2​,…,αn​ are dependent (or independent) instead of saying $S$S is dependent (or independent).

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

Theorem 4. Let $V$V be a vector space which is spanned by a finite set of vectors ${\beta}_1,{\beta}_2,\dots,{\beta}_m$β1​,β2​,…,βm​. Then any independent set of vectors in $V$V is finite and contains no more than $m$m elements.
Corollary 1. If $V$V is a finite-dimensional vector space, then any two bases of $V$V have the same (finite) number of elements.
Corollary 2. Let $V$V be a finite-dimensional vector space and let $n=\dim V$n=dimV. Then
( a ) any sunset of $V$V which contains more than $n$n vectors is linearly dependent;
( b ) no subset of $V$V which contains fewer than $n$n vectors can span $V$V.

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

Theorem 6. If $W_1$W1​ and $W_2$W2​ are finite-dimensional subspaces of a vector space $V$V, then $W_1+W_2$W1​+W2​ is finite-dimensional and
$\dim W_1+\dim W_2=\dim(W_1\cap W_2)+\dim(W_1+W_2)$dimW1​+dimW2​=dim(W1​∩W2​)+dim(W1​+W2​)

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

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

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

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

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

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

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