Question

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

Answer #1

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...

Write each vector as a linear combination of the vectors in
S. (Use s1 and s2, respectively, for
the vectors in the set. If not possible, enter IMPOSSIBLE.)
S = {(1, 2, −2), (2, −1, 1)}
(a) z = (−5, −5,
5)
z = ?
(b) v = (−2, −6,
6)
v = ?
(c) w = (−1, −17,
17)
w = ?
Show that the set is linearly dependent by finding a nontrivial
linear combination of vectors in the set whose sum...

(a) Consider x^2 + 7x + 15 = f(x) and e^x = g(x) which are
vectors of F(R, R) with the usual addition and scalar
multiplication. Are these functions linearly independent?
(b) Let S be a finite set of linearly independent vectors {u1,
u2, · · · , un} over the field Z2. How many vectors are in
Span(S)?
(c) Is it possible to find three linearly dependent vectors in
R^3 such that any two of the three are not...

Using a step by step proof format
Please prove:
Given that x is a vector in the span of V, where V is a linearly
independent set of vectors, show that there is ONLY ONE linear
combination of the vectors in V that yields x. (Hint: to show that
something is unique, assume that there is more than one such thing
and show that this leads to a contradiction)

3. Which of the following sets spans P2(R)?
(a) {1 + x, 2 + 2x 2}
(b) {2, 1 + x + x 2 , 3 + 2x + 2x 2}
(c) {1 + x, 1 + x 2 , x + x 2 , 1 + x + x 2}
4. Consider the vector space W = {(a, b) ∈ R 2 | b > 0} with
defined by (a, b) ⊕ (c, d) = (ad + bc, bd)...

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 X be a real vector space. Suppose {⃗v1,⃗v2,⃗v3} ⊂ X is a
linearly independent set, and suppose {w⃗1,w⃗2,w⃗3} ⊂ X is a
linearly dependent set. Define V = span{⃗v1,⃗v2,⃗v3} and W =
span{w⃗1,w⃗2,w⃗3}.
(a) Is there a linear transformation P : V → W such that P(⃗vi)
= w⃗i for i = 1, 2, 3?
(b) Is there a linear transformation Q : W → V such that Q(w⃗i)
= ⃗vi for i = 1, 2, 3?
Hint: the...

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...

1) Let S={v_1, ..., v_p} be a
linearly dependent set of vectors in R^n. Explain why it is the
case that if v_1=0, then
v_1 is a trivial combination of the other vectors
in S.
2) Let S={(0,1,2),(8,8,16),(1,1,2)} be a set of column vectors
in R^3. This set is linearly dependent. Label each vector in S with
one of v_1, v_2, v_3 and find constants, c_1, c_2,
c_3 such that c_1v_1+ c_2v_2+
c_3v_3=0. Further, identify the
value j and v_j...

Determine if the first vector is a linear combination of the
other two vectors. Show algebraically how you found your
answer.
2x^2 + 2x + 1, -x^2 + 2x + 1, -2x^2 + 2x + 1 in P2 (P subscript
2) (R).

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