Question

Give an counter example or explain why those are false

a) every linearly independent subset of a vector space V is a basis
for V

b) If S is a finite set of vectors of a vector space V and v
⊄span{S}, then S U{v} is linearly independent

c) Given two sets of vectors S1 and S2, if span(S1) =Span(S2), then
S1=S2

d) Every linearly dependent set contains the zero vector

Answer #1

For each of the following statements, say whether the statement
is true or false.
(a) If S⊆T are sets of vectors, then span(S)⊆span(T).
(b) If S⊆T are sets of vectors, and S is linearly independent,
then so is T.
(c) Every set of vectors is a subset of a basis.
(d) If S is a linearly independent set of vectors, and u is a
vector not in the span of S, then S∪{u} is linearly
independent.
(e) In a finite-dimensional...

True or False. Explain.
Every subset of a linearly independent set is linearly
independent.

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}...

Mark the following as true or false, as the case may be. If a
statement is true, then prove it. If a statement is false, then
provide a counter-example.
a) A set containing a single vector is linearly independent
b) The set of vectors {v, kv} is linearly dependent for every
scalar k
c) every linearly dependent set contains the zero vector
d) The functions f1 and f2 are linearly
dependent is there is a real number x, so that...

If v1 and v2 are linearly independent vectors in vector space V,
and u1, u2, and u3 are each a linear combination of them, prove
that {u1, u2, u3} is linearly dependent.
Do NOT use the theorem which states, " If S = { v 1 , v 2 , . . . ,
v n } is a basis for a vector space V, then every set
containing
more than n vectors in V is linearly dependent."
Prove without...

Answer the following T for True and F for False:
__ A vector space must have an infinite number of vectors to be
a vector space.
__ The dimension of a vector space is the number of linearly
independent vectors contained in the vector space.
__ If a set of vectors is not linearly independent, the set is
linearly dependent.
__ Adding the zero vector to a set of linearly independent
vectors makes them linearly dependent.

Show that the set is linearly dependent by finding a nontrivial
linear combination of vectors in the set whose sum is the zero
vector. (Use
s1, s2, and s3, respectively,
for the vectors in the set.)
S = {(5, 2), (−1, 1), (2, 0)}
a) (0, 0) =
b) Express the vector s1 in the set as a
linear combination of the vectors s2 and
s3.
s1 =

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...

Find an example of a nonzero, non-Invertible 2x2 matrix A and a
linearly independent set {V,W} of two, distinct
non-zero vectors in R2 such that
{AV,AW} are distinct, nonzero and
linearly dependent. verify the matrix A in non-invertible, verify
the set {V,W} is linearly independent and verify
the set {AV,AW} is linearly
dependent

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

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