Suppose v1, v2, . . . , vn is linearly independent in V and w ∈...
Suppose v1, v2, . . . , vn is linearly independent in V and w ∈
V . Show that v1, v2, . . . , vn, w is linearly independent if and
only if w ∈/ Span(v1, v2, . . . , vn).
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)
A. Suppose that v1, v2, v3 are linearly independant and
w1=v1+v2, w2=v2-v3, w3= v2+v3. Determine whether...
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)
Suppose that {v1, v2, u} and {v1, v2, w}, are both bases for
some subspace V...
Suppose that {v1, v2, u} and {v1, v2, w}, are both bases for
some subspace V ⊆ R n .
(a) Demonstrate with an example that it is possible for {u, w}
to be linearly independent.
(b) Assuming now that {u, w} is linearly independent, decide
whether {v1, u, w} is necessarily a basis for V , giving
justification for your answer.
For both parts it isn’t sufficient just to give answers, you
must give convincing arguments, making appropriate use...
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 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...
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?
Let A be an m × n matrix such that ker(A) = {⃗0} and let
⃗v1,⃗v2,...,⃗vq...
Let A be an m × n matrix such that ker(A) = {⃗0} and let
⃗v1,⃗v2,...,⃗vq be linearly independent vectors in Rn. Show that
A⃗v1, A⃗v2, . . . , A⃗vq are linearly independent vectors in
Rm.
Prove that
Let S={v1,v2,v3} be a linearly indepedent set of vectors om a
vector space V....
Prove that
Let S={v1,v2,v3} be a linearly indepedent set of vectors om a
vector space V. Then so are
{v1},{v2},{v3},{v1,v2},{v1,v3},{v2,v3}