Question

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.

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...

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...

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...

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

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...

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.

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

7. Answer the following questions true or false and provide an
explanation. • If you think the statement is true, refer to a
definition or theorem. • If false, give a counter-example to show
that the statement is not true for all cases.
(a) Let A be a 3 × 4 matrix. If A has a pivot on every row then
the equation Ax = b has a unique solution for all b in R^3 .
(b) If the augmented...

Consider the following vectors
{1 -x, 1+ x, x-2}
A. test or refute if the vector set is linearly
independent
B. build a linearly independent set of dimension 1, describe
the shape of the generated space
C. build a linearly independent set of dimension 2, describe
the shape of the generated space
D. for the base you chose in part c, find a linear combination
for p(x)=7 - x

Let V be a vector space: d) Suppose that V is
finite-dimensional, and let S be a set of inner products on V that
is (when viewed as a subset of B(V)) linearly independent. Prove
that S must be finite
e) Exhibit an infinite linearly independent set of inner
products on R(x), the vector space of all polynomials with real
coefficients.

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