Question

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>

Answer #1

Let v = (v_{1}, · · · , v_{n}) be a fixed
vector? R^{n} and let w = (w_{1}, · · · ,
w_{n}) ? R^{n} be an arbitrary vector. Further, let
us define T: R^{n} ? R by T(w) = < v,w >. We will
show that T is a linear transformation.

Let w = (w_{1}, · · · , w_{n}) and u =
(u_{1}, · · · , u_{n}) be 2 arbitrary vectors ?
R^{n} and let k be an arbitrary scalar. Then T(w+u)=
<v,w+u> = v_{1}(w_{1}+u_{1}) + · · ·
+ v_{n} (w_{n}+u_{n}) =
v_{1}w_{1} + · · · + v_{n}w_{n} +
v_{1} u_{1} + · · · + v_{n} u_{n} =
< v,w > +< v,u > = T(w)+T(u). This means that T
preserves vector addition. Also, T(kw) =<v,kw > =
v_{1}kw_{1} + · · · + v_{n} kw_{n}=
k(v_{1}w_{1} + · · · + v_{n}w_{n})
= kT(w). This means that T preserves scalar multiplication. Hence T
is a linear transformation.

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.

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 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
{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 V and W be finite-dimensional vector spaces over F, and let
φ : V → W be a linear transformation. Let dim(ker(φ)) = k, dim(V )
= n, and 0 < k < n. A basis of ker(φ), {v1, . . . , vk}, can
be extended to a basis of V , {v1, . . . , vk, vk+1, . . . , vn},
for some vectors vk+1, . . . , vn ∈ V . Prove that...

5. Prove or disprove the following statements.
(a) Let L : V → W be a linear mapping. If {L(~v1), . . . , L(
~vn)} is a basis for W, then {~v1, . . . , ~vn} is a basis for
V.
(b) If V and W are both n-dimensional vector spaces and L : V →
W is a linear mapping, then nullity(L) = 0.
(c) If V is an n-dimensional vector space and L : V →...

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.

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}...

Suppose v1, v2, . . . , vn is linearly independent in V and w ∈
V . Show that v1, v2, . . . , vn, w is linearly independent if and
only if w ∈/ Span(v1, v2, . . . , vn).

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.

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