K2) (a) Show that the vectors (1, 1, 1), (1, 2, 3) and (2, 3, 8) are linearly independent
Answers
Answer:
Last lecture: Examples and the column space of a matrix
Suppose that A is an n × m matrix.
Definition The column space of A is the vector subspace
Col(A) of R
n
which is spanned by the columns of A.
That is, if A =
£
a1, a2, . . . , am
¤
then Col(A) =
Span ¡
a1, a2, . . . , am
¢
.
Linear dependence and independence (chapter. 4)
• If V is any vector space then V = Span(V ).
• Clearly, we can find smaller sets of vectors which
span V .
• This lecture we will use the notions of linear
independence and linear dependence to find the
smallest sets of vectors which span V .
• It turns out that there are many “smallest sets” of
vectors which span V , and that the number of
vectors in these sets is always the same.
This number is the dimension of V .
0-0Linear dependence—motivation Let lecture we saw that
the two sets of vectors n·
1
2
3
¸
,
·
3
5
7
¸
,
·
5
9
13 ¸
o
and
n·
1
2
3
¸
,
·
3
5
7
¸
,
·
0
1
2
¸
o
do not span R
3
.
• The problem is that
·
5
9
13 ¸
= 2·
1
2
3
¸
+
·
3
5
7
¸
and
·
0
1
2
¸
= 3·
1
2
3
¸
−
·
3
5
7
¸
.
• Therefore,
Span ³
·
1
2
3
¸
,
·
3
5
7
¸
,
·
5
9
13 ¸
´
= Span ³
·
1
2
3
¸
,
·
3
5
7
¸
´
and
Span ³
·
1
2
3
¸
,
·
3
5
7
¸
,
·
0
1
2
¸
´
= Span ³
·
1
2
3
¸
,
·
3
5
7
¸
´
.
• Notice that we can rewrite the two equations
above in the following form:
2
·
1
2
3
¸
+
·
3
5
7
¸
−
·
5
9
13 ¸
=
·
0
0
0
¸
and
3
·
1
2
3
¸
−
·
3
5
7
¸
−
·
0
1
2
¸
=
·
0
0
0
¸
0-1This is the key observation about spanning sets.
Definition
Suppose that V is a vector space and that x1, x2, . . . , xk
are vectors in V .
The set of vectors {x1, x2, . . . , xk
} is linearly dependent
if
r1
x1 + r2
x2 + · · · + rk
xk = 0
for some r1, r2, . . . , rk ∈ R where at least one of r1, r2, . . . , rk
is non–zero.
Example
2
·
1
2
3
¸
+
·
3
5
7
¸
−
·
5
9
13 ¸
=
·
0
0
0
¸
and
3
·
1
2
3
¸
−
·
3
5
7
¸
−
·
0
1
2
¸
=
·
0
0
0
¸
So the two sets of vectors n·
5
9
13 ¸
,
·
1
2
3
¸
,
·
3
5
7
¸
o
and
n·
0
1
2
¸
,
·
1
2
3
¸
,
·
3
5
7
¸
o
are linearly dependent.
Question Suppose that x, y ∈ V . When are x and y
linearly dependent?
0-2Question What do linearly dependent vectors look like
in R
2
and R
3
?
Example
Let x =
"
1
2
3
#
y =
"
3
2
1
#
and z =
"
0
4
8
#
. Is {x1, x2, x3
}
linearly dependent?
We have to determine whether or not we can find real
numbers r, s,t, which are not all zero, such that
rx + sy + tz = 0.
To find all possible r, s,t we have to solve the augmented
matrix equation:
"
1 3 0 0
2 2 4 0
3 1 8 0
#
R2:=R2−2R1
−−−−−−−−→
R3:=R3−3R1
"
1 3 0 0
0 −4 4 0
0 −8 8 0
#
R3:=R3−2R2
−−−−−−−−→
"
1 3 0 0
0 −4 4 0
0 0 0 0
#
So this set of equations has a non–zero solution.
Therefore, {x, y, z} is a linearly dependentset of vectors.
Therefore, {cos x,sin x, x} is linearly independent.
Step-by-step explanation:
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