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...
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?
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)
if
{Av1,Av2,...,
Avk} is linearly dependent set
of vectors in Rn and A is an nxn...
if
{Av1,Av2,...,
Avk} is linearly dependent set
of vectors in Rn and A is an nxn invertible
matrix, the
{v1,v2,...vk}
is also a linearly dependent set of vectors in Rn
1. Prove that if {⃗v1, ⃗v2, ⃗v3} is a linear dependent set of
vectors in V...
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.
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 ?
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...