If v1 and v2 are linearly independent vectors in vector space V,
and u1, u2, and...
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...
If S=(v1,v2,v3,v4)
is a linearly independent sequence of vectors in Rn
then
A) n = 4...
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)
Let
{V1, V2,...,Vn} be a linearly independent set of vectors choosen
from vector space V. Define...
Let
{V1, V2,...,Vn} be a linearly independent set of vectors choosen
from vector space V. Define w1=V1, w2= v1+v2, w3=v1+ v2+v3,...,
wn=v1+v2+v3+...+vn.
(a) Show that {w1, w2, w3...,wn} is a linearly independent
set.
(b) Can you include that {w1,w2,...,wn} is a basis for V? Why
or why not?
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]...
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?
Determine if the vectors v1= (3, 0, -3, 6),
v2 = ( 0, 2, 3, 1),...
Determine if the vectors v1= (3, 0, -3, 6),
v2 = ( 0, 2, 3, 1), and v3 = (0, -2, 2, 0 )
form a linearly dependent set in R 4. Is it a basis of
R4 ?
Determine if vectors are linearly dependent or
independent:
1. (1,2), (-1,-3)
2. (2,-1,4),(4,-2,7),(1,5,8)
3. (-3,4,2),(7,-1,3),(1.1.8)
Determine if vectors are linearly dependent or
independent:
1. (1,2), (-1,-3)
2. (2,-1,4),(4,-2,7),(1,5,8)
3. (-3,4,2),(7,-1,3),(1.1.8)
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],...
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...
Let X be a real vector space. Suppose {⃗v1,⃗v2,⃗v3} ⊂ X is a
linearly independent set,...
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...