documentclass[11pt]article usepackageamssymb,amsmath,latexsym usepackagegraphicx par newedcommandbfibfseriesitshape par textwidth 6.5 truein oddsidemargin 0 truein evensidemargin -0.50 truein topmargin -.5 truein textheight 8.5in par newedtheoremthmTheorem[section] newedtheoremprop[thm]Proposition newedtheoremlem[thm]Lemma newedtheoremcor[thm]Corollary newedtheoremdfn[thm]Definition par newedcommandintprod[2]#1hook#2 newedcommandhookmathbinhboxvrule height .5pt width 3.5pt depth 0pt vrule height 6pt width .5pt depth 0pt par makeatletter addtoresetfiguresubsection addtoresettablesubsection addtoresetequationsubsection makeatother par makeatletter par newskip centering centering = 0pt plus 1000pt par eqnsel= par makeatother par begindocument par titleReview: Vector Spaces par authorCDS201 Lecturer: W.S. Koon datesmall Fall, 1998 maketitle par sectionReal and Complex Vector Spaces paragraphDefinition: paragraphExamples: beginenumerate item $R^n$ item $C^n$ item $C[0,1]$ item $L^2[0,1]$ item matrices item polynomials endenumerate par sectionSubspace paragraphDefinition: paragraphExamples: beginenumerate item Let $V$ be any vector space, $V$ is a subspace of $V$ and $\{0\}$ is a subspace of $V$. item $V=R^3$, $W=\{(a,b.c)in R^3: c=0\}$. item $V=$ space of $ntimes n$ matrices, $Wsubset V$ is the subset of symmetric matrices. item $V$ is the set of all polynomials of degree $n$ with real coefficients, $W$ is the subset of $V$ consisting of polynomials of degree $m leq n$. item $U, W$ subspaces of $V$, $Ucap W$ is a subspace of $V$. endenumerate par sectionLinear Combinations and Spans paragraphDefinition: Let $V$ be a vector space, $v_1, ldots , v_m in V$. Any vector in $V$ of the form $a_1v_1 + cdots + a_mv_m$ is called a it linear combination of $v_1, ldots , v_m$. $a_1, ldots ,a_m$ are called the coefficients of the linear combinations. par paragraphDefinition: Let $S$ be a subset of a vector space $V$. $L(S)$ denotes the set of all linear combinations of vectors in $S$. It can be shown that $L(S)$ is a subspace, and it is said to be the subspace it spanned or generated by $S$. par sectionDirect Sum paragraphDefinition: Suppose $U$ and $W$ are subspace of a vector space $V$. Then the sum of $U$ and $W$, denoted $U+W$, consists of all sums $u+w$ where $uin U$, $win W$, i.e.

$$U+W=\{u+w\vert uin U, win W \}$$

par paragraphDefinition: $V$ is said to be the it direct sum of $U$ and $W$ if for any $vin V$ there exists a unique $uin U, win W$ such that $v=u+w$. $V=Uoplus W$. par paragraphExamples: $V=R^3$ beginenumerate item $U=\{(a,b,0)\vert a,bin R\}, W=\{(0,a,b)\vert a,bin R\}$. Notice that while $V=U+W$, $V$ is not the direct sum of $U$ and $W$. item $U=\{(a,b,0)\vert a,bin R\}, W=\{(0,0,a)\vert ain R\}$. $V=Uoplus W$. endenumerate par begintheorem_type[prop][thm][][][][] Suppose $U$ and $W$ are subspaces of $V$. Then $U+W$ is also a subspace of $V$.endtheorem_type par begintheorem_type[thm][thm][section][][][] Let $U, W$ be subspaces of a vector space $V$. Then $V$ is a direct sum of $U$ and $W$ if and only if beginenumerate item $V=U+W$ item $Ucap W =\{0\}$. endenumerateendtheorem_type vspace2in par sectionLinear Independence and Dependence paragraphDefinition: Let $V$ be a vector space and let $v_1, ldots , v_m in V$. The set $\{v_1, ldots , v_m\}$ is said to be it linearly independent if $a^1v_1 + cdots + a^mv_m =0$ implies $a^1= cdots = a^m=0$. Otherwise, they are said to be it linearly dependent. par paragraphExample: Linear dependence in $R^n$ Problem of linear dependence is related to a nontrivial solution of a homogeneous system of linear equations. par begintheorem_type[prop][thm][][][][] Let $\{v_1, ldots , v_m\}$ be a collection of nonzero vectors. Then this set of vectors is linearly dependent if and only if one of the vectors can be expressed as a linear combination of the others.endtheorem_type vspace1in par sectionBases, Coordinates and Dimension paragraphDefinition: Let $\{e_1, ldots , e_n\}$ be a set of linearly independent vectors in a vector space $V$. Then $\{e_1, ldots , e_n\}$ is said to be a it basis for $V$ if any $vin V$ can be expressed as a linear combination of $\{e_1, ldots , e_n\}$, i.e. $v=a^1e_1 +cdots + a^ne_n$. The numbers $a^1, ldots , a^n$ are called the components or it coordinates of the vector $v$ with respect to the basis $\{e_1, ldots , e_n\}$. The components are unique for a given basis. par paragraphDefinition: A vector space $V$ is said to have dimension $n$ if it has $n$ linearly independent vectors while every set of $n+1$ vectors are linearly dependent. par begintheorem_type[thm][thm][section][][][] In a vector space $V$ of dimension $n$ there exists a basis consisting of $n$ independent vectors. Moreover, any set of $n$ linearly independent vectors in $V$ is a basis for $V$.endtheorem_type par begintheorem_type[thm][thm][section][][][] If there is a basis in $V$, then the dimension of $V$ equals the number of basis vectors.endtheorem_type vspace5in par sectionUseful Theorems Concerning Subspaces and Bases begintheorem_type[thm][thm][section][][][] Suppose $W$ is a subspace of the vector space $V$ where dim $V=n$ and let $\{w_1,ldots , w_l\}$ be a basis for $W$. Then there exists vectors $\{e_{l+1}, ldots , e_n\}$ such that $\{w_1, ldots ,w_l, e_{l+1}, ldots , e_n\}$ is a basis of $V$.endtheorem_type par begintheorem_type[thm][thm][section][][][] Suppose $U$ and $W$ are subspaces of a finite dimensional vector space $V$. Then dim $(U+W)=$ dim $U$ + dim $W$ $-$ dim $(Ucap W)$.endtheorem_type par begintheorem_type[thm][thm][section][][][] Let $V$ be a vector space of dimension $n$ and suppose $Wsubset V$ is a subspace of $V$. Then there exists a subspace $Usubset V$ such that $V=Uoplus W$.endtheorem_type enddocument