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.
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)
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}
Verify that the following vectors form an orthogonal list:
v1=1, 1, 2 v2= 2,0,-1] v3=1,...
Verify that the following vectors form an orthogonal list:
v1=1, 1, 2 v2= 2,0,-1] v3=1, −5, 2
Problem 6.4 What does it mean to say that a set of vectors {v1,
v2, ....
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, 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 is the set of vectors in R^4 (S= {v1, v2, v3, v4, v5})
where,...
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).
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...
Suppose ⃗v1,⃗v2,⃗v3,⃗v4 ∈ R3. Let V = {⃗v1,⃗v2,⃗v3,⃗v4} and let
X = [⃗v1|⃗v2|⃗v3|⃗v4] be the matrix...
Suppose ⃗v1,⃗v2,⃗v3,⃗v4 ∈ R3. Let V = {⃗v1,⃗v2,⃗v3,⃗v4} and let
X = [⃗v1|⃗v2|⃗v3|⃗v4] be the matrix whose columns are
⃗v1,⃗v2,⃗v3,⃗v4. Suppose further that every subset Y ⊂ V of size
two is linearly independent. Explain what form(s) rref(X), the
reduced row echelon form of X, must take in this case. Hint: you
won’t be able to pin down exact numbers for every entry of rref(X),
but you might know things like whether the entry can be zero or
not, etc.