Question

Let V be a vector space of dimension n > 0, show that

(a) Any set of n linearly independent vectors in V forms a
basis.

(b) Any set of n vectors that span V forms a basis.

Answer #1

(a) Any set of n linearly independent vectors in V forms a basis.

Suppose that S = {v1,v2,…,vn} are n linearly independent
vectors. Take any non-zero vector v∈V. As V is of dimension n, the
vectors v,v1,…,vn satisfy a linear dependency.

where not all the coefficients α,α1,…,αn are 0. In fact, α≠0 as S
is linearly independent. Therefore, one can write

which proves that S is a basis for V

(b) Any set of n vectors that span V forms a basis

Suppose that S = {v1,v2,…,vn} span V.

If vectors are not linearly independent.

v1 can be written as

which shows dimension of S is having dimension less than n.
(given dim(V) = n) which is clearly false. so, the vectors of set S
are linearly independent.

Hence from (a) , any set of n linearly independent vectors forms
basis. So, S forms basis.

hence proved.

Let V be a vector space with dimV = n.
Show that : Any spanning set for V consisting of exactly n
vectors is a basis for V.

Suppose we have a vector space V of dimension n. Let R be a
linearly independent set with order n−2. Let S be a spanning set
with order n+ 2. Outline a strategy to extend R to a basis for V.
Outline a strategy to pare down S to a basis for V .

Let V be an n-dimensional vector space and W a vector
space that is isomorphic to V. Prove that W is also
n-dimensional. Give a clear, detailed, step-by-step
argument using the definitions of "dimension" and "isomorphic"
the Definiton of isomorphic: Let V be an
n-dimensional vector space and W a vector space that is
isomorphic to V. Prove that W is also n-dimensional. Give
a clear, detailed, step-by-step argument using the definitions of
"dimension" and "isomorphic"
The Definition of dimenion: the...

Let S be a set in a vector space V and v any vector. Prove that
span(S) = span(S ∪ {v}) if and only if v ∈ span(S).

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?

Definition. Let S ⊂ V be a subset of a vector space. The span of
S, span(S), is the set of all finite
linear combinations of vectors in S. In set notation,
span(S) = {v ∈ V : there exist v1, . . . , vk ∈ S and a1, . . . ,
ak ∈ F such that v = a1v1 + . . . + akvk} .
Note that this generalizes the notion of the span of a...

Let T be a 1-1 linear transformation from a vector space V to a
vector space W. If the vectors u,
v and w are linearly independent
in V, prove that T(u), T(v),
T(w) are linearly independent in W

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.

Prove that if a vector space V has a basis consisting of n
vectors, then any spanning set must contain at least n vectors.

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