Definition. Let S ⊂ V be a subset of a vector space. The span of
S,...
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...
For a nonempty subset S of a vector space V , define span(S) as
the set...
For a nonempty subset S of a vector space V , define span(S) as
the set of all linear combinations of vectors in S.
(a) Prove that span(S) is a subspace of V .
(b) Prove that span(S) is the intersection of all subspaces that
contain S, and con- clude that span(S) is the smallest subspace
containing S. Hint: let W be the intersection of all subspaces
containing S and show W = span(S).
(c) What is the smallest subspace...
Let v1=(0,1,2,3),v2=(1,0,-1,0),v3=(0,4,-1,2), and v4=(0,5,1,5).
Let S=(v1,v2,v3,v4)
(1)find a basis for span(S)
(2)is the vector e1=(1,0,0,0) in...
Let v1=(0,1,2,3),v2=(1,0,-1,0),v3=(0,4,-1,2), and v4=(0,5,1,5).
Let S=(v1,v2,v3,v4)
(1)find a basis for span(S)
(2)is the vector e1=(1,0,0,0) in the span of S? Why?
Let T(x,y,z) = (13x − 9y + 4z,6x + 5y − 3z) and v = (1,−2,1)....
Let T(x,y,z) = (13x − 9y + 4z,6x + 5y − 3z) and v = (1,−2,1).
Find the standard matrix for the linear transformation T and use it
to find the image of the vector v (that is, use it to find
T(v)).
5.
Let S be the set of all polynomials p(x) of degree ≤ 4 such
that...
5.
Let S be the set of all polynomials p(x) of degree ≤ 4 such
that
p(-1)=0.
(a) Prove that S is a subspace of the vector space of all
polynomials.
(b) Find a basis for S.
(c) What is the dimension of S?
6.
Let ? ⊆ R! be the span of ?1 = (2,1,0,-1), ?2
=(1,2,-6,1),
?3 = (1,0,2,-1) and ? ⊆ R! be the span of ?1 =(1,1,-2,0), ?2
=(3,1,2,-2). Prove that V=W.
4. Prove the Following:
a. Prove that if V is a vector space with subspace 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 S = {(5,6,7), (6,7,8), (7,8,9)}.
(a) What equations does span(S) obey? In other words,
identify...
Let S = {(5,6,7), (6,7,8), (7,8,9)}.
(a) What equations does span(S) obey? In other words,
identify span(S) using a minimal set of equations.
(b) Is v → = (3, 5, 7) in span(S)? Prove your answer.
Complete the proof
Let V be a nontrivial vector space which has a spanning set
{xi}...
Complete the proof
Let V be a nontrivial vector space which has a spanning set
{xi} ki=1. Then there is a subset
of {xi} ki=1 which is a basis for
V.
Proof. We will divide the set {xi}
ki=1 into two sets, which we will call
good and bad. If x1 ≠ 0, then we label
x1 as good and if it is zero, we label it as
bad. For each i ≥ 2, if xi ∉
span{x1, . ....