Theorem. For any set of vectors \(\displaystyle{S}={\left\lbrace{v}{1},…,{v}{n}\right\rbrace}\) in a vector space V, span (S) is a subspace of V.
Proof. Let \(\displaystyle{u},{w}∈{s}{p}{a}{n}{\left({S}\right)},{k}∈{F}.\) Then there exist \(\displaystyle{c}{1},…,{c}{n}∈{R},…,{c}{n}∈{R}{\quad\text{and}\quad}{k}{1},…,{k}{n}∈{F}{k}∈{F}\) such that
\(\displaystyle{u}={c}{1}{v}{1}+{c}{2}{v}{2}+\ldots+{c}{n}{v}{n}\)
and
\(\displaystyle{w}={k}{1}{v}{1}+{k}{2}{v}{2}+\ldots+{k}{n}{v}{n}\)
Note that
\(\displaystyle{u}+{w}={\left({c}{1}{v}{1}+{c}{2}{v}{2}+\ldots+{c}{n}{v}{n}\right)}+{\left({k}{1}{v}{1}+{k}{2}{v}{2}+\ldots+{k}{n}{v}{n}\right)}={\left({c}{1}+{k}{1}\right)}{v}{1}+{\left({c}{2}+{k}{2}\right)}{v}{2}+\ldots+{\left({c}{n}+{k}{n}\right)}{v}{n}∈{s}{p}{a}{n}{\left({S}\right)}\)
Thus span (S) is closed under addition. M1)
\(ku=k(c_1v_1+c_2v_2+\dots+c_nv_n) =k(c_1v_1)+k(c_2v_2)+\dots+k(c_nv_n) =(kc_1)v_1+(kc_2)v_2+\dots+(kc_n)v_n \in span(S)\)
This hosws that span (S) is closed under scalar multiplication. Hence , span (S) is a subspace of V.
Let S be a subset of an F-vector space V. Show that Span(S) is a subspace of V.
Answers (1)
Relevant Questions
A. V=\(P_5\), and S is the subset of \(P_5\) consisting of those polynomials satisfying p(1)>p(0).
B. \(V=R_3\), and S is the set of vectors \((x_1,x_2,x_3)\) in V satisfying \(x_1-6x_2+x_3=5\).
C. \(V=R^n\), and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=\(C^2(I)\), and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.
E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.
F. V=\(P_n\), and S is the subset of \(P_n\) consisting of those polynomials satisfying p(0)=0.
G. \(V=M_n(R)\), and S is the subset of all symmetric matrices
Consider \(\displaystyle{V}={s}{p}{a}{n}{\left\lbrace{\cos{{\left({x}\right)}}},{\sin{{\left({x}\right)}}}\right\rbrace}\) a subspace of the vector space of continuous functions and a linear transformation \(\displaystyle{T}:{V}\rightarrow{V}\) where \(\displaystyle{T}{\left({f}\right)}={f{{\left({0}\right)}}}\times{\cos{{\left({x}\right)}}}−{f{{\left(π{2}\right)}}}\times{\sin{{\left({x}\right)}}}.\) Find the matrix of T with respect to the basis \(\displaystyle{\left\lbrace{\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}},{\cos{{\left({x}\right)}}}−{\sin{{\left({x}\right)}}}\right\rbrace}\) and determine if T is an isomorphism.
Let U and W be vector spaces over a field K. Let V be the set of ordered pairs (u,w) where \(u \in U\) and \(w \in W\). Show that V is a vector space over K with addition in V and scalar multiplication on V defined by
\((u,w)+(u',w')=(u+u',w+w')\ and\ k(u,w)=(ku,kw)\)
(This space V is called the external direct product of U and W.)
Let V be the vector space of real 2 x 2 matrices with inner product
\((A|B) = tr(B^tA)\).
Let U be the subspace of V consisting of the symmetric matrices. Find an orthogonal basis for \(U^\perp\) where \(U^{\perp}\left\{A \in V |(A|B)=0 \forall B \in U \right\}\)
