Question

1. Let v1,…,vn be a basis of a vector space V. Show that

(a) for any non-zero λ1,…,λn∈R, λ1v1,…,λnvn is also a basis of V.

(b) Let ui=v1+⋯+vi, 1≤i≤n. Show that u1,…,un is a basis of V.

Answer #1

† Let β={v1,v2,…,vn} be a basis for a vector space V
and T:V→V be a linear transformation. Prove that [T]β is upper
triangular if and only if T(vj)∈span({v1,v2,…,vj}) j=1,2,…,n. Visit
goo.gl/k9ZrQb for a solution.

Let W be an inner product space and v1,...,vn a basis of V. Show
that〈S, T 〉 = 〈Sv1, T v1〉 + . . . + 〈Svn, T vn〉
for S,T ∈ L(V,W) is an inner product on L(V,W).
Let S ∈ L(R^2) be given by S(x1, x2) = (x1 + x2, x2) and let I ∈
L(R^2) be the identity operator. Using the inner product defined in
problem 1 for the standard basis and the dot product, compute 〈S,...

Let V be a vector space and let v1,v2,...,vn be elements of V .
Let W = span(v1,...,vn). Assume v ∈ V and ˆ v ∈ V but v / ∈ W and ˆ
v / ∈ W. Deﬁne W1 = span(v1,...,vn,v) and W2 = span(v1,...,vn, ˆ
v). Prove that either W1 = W2 or W1 ∩W2 = W.

4. Prove the Following:
a. Prove that if V is a vector space with subspace W ⊂ V, and if
U ⊂ W is a subspace of the vector space W, then U is also a
subspace of V
b. Given span of a finite collection of vectors {v1, . . . , vn}
⊂ V as follows:
Span(v1, . . . , vn) := {a1v1 + · · · + anvn : ai are scalars in
the scalar field}...

Let
{V1, V2,...,Vn} be a linearly independent set of vectors choosen
from vector space V. Define w1=V1, w2= v1+v2, w3=v1+ v2+v3,...,
wn=v1+v2+v3+...+vn.
(a) Show that {w1, w2, w3...,wn} is a linearly independent
set.
(b) Can you include that {w1,w2,...,wn} is a basis for V? Why
or why not?

Let B={u1,...un} be an orthonormal basis for inner product space
V and v b any vector in V. Prove that v =c1u1 + c2u2 +....+cnun
where c1=<v,u1>, c2=<v,u2>,...,cn=<v,un>

Let v = (v1, · · · , vn), w = (w1, · · · , wn) ? R^n and let
<v, w> denote the inner product on R n given by <v, w>=
v1w1 + · · · + vnwn. Prove that for any linear transformation T :
R^n ? R, there exists a fixed vector v ? R^n such that T(w) =
<v, w>

Suppose the vectors v1, v2, . . . , vp span a vector space V
.
(1) Show that for each i = 1, . . . , p, vi belongs to V ;
(2) Show that given any vector u ∈ V , v1, v2, . . . , vp, u also
span V

let T:V to W be a linear transdormation of vector
space V and W and let B=(v1,v2,...,vn) be a basis for V. Show that
if (Tv1,Tv2,...,Tvn) is linearly independent, thenT is
injecfive.

T/ F : Let V be an inner product space with orthogonal basis B =
{v1, . . . , vn}. Let [v]B = (1, 2, 2, 0, . . . , 0). Then ||v|| =
3.
The ans is F , but I don't understand why. Please explain.

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