Question

Determine all real numbers a for which the vectors

v1 = (1,−1,1,a,2)

v2 = (−1,0,0,1,0)

v3 = (1,2,a + 1,1,0)

v4 = (2,0,a + 3,2a + 3,4)

make a linearly independent set. For which values of a does the set contain at least three linearly independent vectors?

Answer #1

If S is the set of vectors in R^4 (S= {v1, v2, v3, v4, v5})
where, v1 = (1,2,-1,1), v2 = (-3,0,-4,3), v3 = (2,1,1,-1), v4 =
(-3,3,-9,-6), v5 = (3,9,7,-6)
Find a subset of S that is a basis for the span(S).

A. Suppose that v1, v2, v3 are linearly independant and
w1=v1+v2, w2=v2-v3, w3= v2+v3. Determine whether w1, w2, w3 are
linear independent or linear deppendent.
B. Find the largest possible number of independent vectors
among:
v1=(1,-1,0,0), v2=(1,0,-1,0), v3=(1,0,0,-1), v4=(0,1,-1,0),
v5=(0,1,0,-1), v6=(0,0,1,-1)

If S=(v1,v2,v3,v4)
is a linearly independent sequence of vectors in Rn
then
A) n = 4
B) The matrix ( v1 v2 v3
v4) has a unique pivot column.
C) S is a basis for
Span(v1,v2,v3,v4)

Determine whether the given vectors span R3
V1=(-1,5,2), V2=(3,1,1), V3=(2,0,-2), V4=(4,1,0)

1. Prove that if {⃗v1, ⃗v2, ⃗v3} is a linear dependent set of
vectors in V , and if ⃗v4 ∈ V , then {⃗v1, ⃗v2, ⃗v3, ⃗v4} is
also a linear dependent set of vectors in V .
2. Prove that if {⃗v1,⃗v2,...,⃗vr} is a linear dependent set of
vectors in V, and if⃗ vr + 1 ,⃗vr+2,...,⃗vn ∈V, then
{⃗v1,⃗v2,...,⃗vn} is also a linear dependent set of vectors in
V.

Consider four vectors v1 = [1,1,1,1], v2 = [-1,0,1,2], v3 =
[a,1,0,b], and v4 = [3,2,a+b,0], where a and b are parameters. Find
all conditions on the values of a and b (if any) for which:
1. The number of linearly independent vectors in this collection
is 1.
2. The number of linearly independent vectors in this collection
is 2.
3. The number of linearly independent vectors in this collection
is 3.
4. The number of linearly independent vectors in...

Problem 6.4 What does it mean to say that a set of vectors {v1,
v2, . . . , vn} is linearly dependent? Given the following vectors
show that {v1, v2, v3, v4} is linearly dependent. Is it possible to
express v4 as a linear combination of the other vectors? If so, do
this. If not, explain why not. What about the vector v3? Anthony,
Martin. Linear Algebra: Concepts and Methods (p. 206). Cambridge
University Press. Kindle Edition.

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?

Are vectors [1,0,0,2,1], [0,1,0,1,−4], and [0,0,1,−1,−1], and
[3,1,5,2,−6] linearly independent?
Are vectors v1=[−16,1,−39], v2=[2,6,3] and v3=[3,1,7] linearly
independent?

Verify that the following vectors form an orthogonal list:
v1=1, 1, 2 v2= 2,0,-1] v3=1, −5, 2

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