Real Vectorspaces is Subspaces Proof (Sketch)
.
1 = 3
Definition. A vectorspace (V , + , ,
)
, over
Let SieS GeR ,
.
Vie [n]. Notice
a field F
,
is an Abelian all cises by axiom 6. ,
under addition equipped with 2 5 , c5 ES by closure
group , , ,
multiplication such that for Note (e,be 5) a cases by , ,
all reV
, ,
and a Bef closure .
I
avev
2
x(pv) (xp)vev = Induction Hypothesis ,
c
,
5 , 0 ...ES
3 x(v0n) =
xv oxieV kn
Y(xop) =
x op = V
s Iv = = lef Then by closure ,
, ,
&
Strat S
.
&
25 1
...
, ,
S
I hus ,
1 33 32 .
Il
Definition .
A subset W&V is a
V if W forms Assume V 2S , /R
of 3 1
subspace , , ,
c
a rectorspace under inherited es bes , ,
es.
operations . Notice,
[83 Axiom I
&
·
I rivial subspace : Let 2
6=
1 .
, ,
W V
Improper subspace =
es bus = S be S .
·
:
, ,
·
Proper subspace : WE
Axiom 2 C
,
= =
C .
,
Propection A novemptysubseta
Note o is inherited
from V
,
so t is
only if associative .
1 w, 2W ,
V
,,, W
2 creW
,
VeeR ,
weW Axiom 3 ,
Let 2
, ==
0 .
05 105 ,
,
= 58 Jes =
.
Proof is
straightforward.
Axiom 4 ,
Let c
,
= 1
, 4
=
Proposition Let SEV ,
(V ,
b
,
%) is a observe , 2
,
5S
S
ventorspace S .
D The following
S
.
= St
=
-
5 =
y
are
equivalent.
Ssubspace of V
.is a Axiom 5 commutativity
I
25 eS c ,R , 5,eS , ,
is inherited from .
V
IS is closed under lines combos I
of 2 vectors ) .
3
VSi ,
Kin eS
, ,
citIR ,
s ,
es
, Span : Linear Independence Exercise Let A = Ex, . .
. 3c V
.
Suppose A is independent .
Definition Let unEV V
v vsp a Prove all subsets of A
.
....., ,
is a
. .
are
Let 2,
,
. .
.,
entR Then < 5 .
, ,
0 ... cUn linearly independent .
is called a liner combination of .
b If A were dependent ,
, In For a , ... .
subset W of V
, would a still hold ?
the Span(W) is the set of all
linear combinations of vectors in .
W
Definition . A basis for vectorspace
Terminology :
W is called a
spanning
V is a linearly independent
set of span(W). spanning set .
Lemma .
In a
vectorspace V the spar
, ,
of any set SSV is a Linear Transformations (CTSI)
subspace .
Definition. Let VW be a
ventorspace .
Definition .
Let VieV for ic[n] ·
If there A function 2: V-W is a
,
exist citR ,
1 :* 0 ,
such that linear transformationif for all
c
,
v, b .. -
bran = 0
,
then V , , . . In are , EV3 CelR ,
linerly dependent. If the only solution 1 ((n ov) L(u)v((v) =
to the above for all
· L(c) cL(v)
equation
=
is
then we say
3
ci 0 =
, , ... Un se
linearly independent. Lemma .
Let F : V -W be a function
Then the V
following an equivalent VCIR V-
, , ,
If is a linear transformation
f(c , , acf(v)
Theme c *) f(t)
2 =
c
N , ,
·
f) Ff(n) =
a Evi] an lim .
dep .
17 Jexo is
a solution to Ac = 5 .
Remark· I distributes over any finite sum
b .
Ev 3 , are independent of the only of rectors from V.
Ac=O is
5
solution to e = .
c .
JeV is in SpanEV3+ iff
= a solution to Ac = b .
Lemma .
Any collection of Kan rectors
in IR" is dependent .
, Example .
F : /" >
- R3 ; (5) +(3) To be in I/V W) , , efocy must be
a liner transformation
(ii) (5)
.
pf .
Let v .
=
82 = <R? coR
.
,
Observe ,
2)(i) (2) 2)(ii) =
S
-
(3(a)(3) Recall D
(For futur use) Definition . For an
(3)(3) 2(v )02(v)
.
-
=
,
nxn matrix M let [Mn , , u
=
,
Mne-man J
that is let in be the nth
,
row of .
M
Next consider
,
((c) =
((c()) (() = Then , define the determinant of M:
det : Mnxn- IR such that,
-
(3) (i) = det(M) det ( +2
det i
, Tn) =
,....
2
,
,
. . .
,x
.
Firti ,
. . .
, In) Getti =
, ...
Ti ...
Mn)
-
c(i) ix]
~
dete , .... Fi , ...i fi .
. .
.. en) -dette ... j.. ... In)
=
, : ...
Thus L is a liner transformation .
I
i *
]
4
dete , . . ., Ki , . . .
. [r) = K .
det[, . .
., en)
5
det(1) = 1 , where is an idembity
Proposition . Let V W
,
be
ventorspaces .
Let matrix
.
2)V W) ,
be the set of all lin. transformation
from V to W .
I(V W) forms ,
a
Vectorspace IR under
over
componentwise
addition.
Recall Week I worksheet problem 1 .
She can show I/V W) ,
is a vsp
by showing that I/V ,W) is a subspace
of F(S) .
Consider z : V-W .
W has Ow ,
since
W is a
vsp , define 7: >Wit Or .
zeI)V , W) ,
and thus I is non
empty -
Take ,R ,
f ze](V W)
, ,
.
Recall function addition is
(f j)(f)
+ =
f(x)oy(v)
.
1 = 3
Definition. A vectorspace (V , + , ,
)
, over
Let SieS GeR ,
.
Vie [n]. Notice
a field F
,
is an Abelian all cises by axiom 6. ,
under addition equipped with 2 5 , c5 ES by closure
group , , ,
multiplication such that for Note (e,be 5) a cases by , ,
all reV
, ,
and a Bef closure .
I
avev
2
x(pv) (xp)vev = Induction Hypothesis ,
c
,
5 , 0 ...ES
3 x(v0n) =
xv oxieV kn
Y(xop) =
x op = V
s Iv = = lef Then by closure ,
, ,
&
Strat S
.
&
25 1
...
, ,
S
I hus ,
1 33 32 .
Il
Definition .
A subset W&V is a
V if W forms Assume V 2S , /R
of 3 1
subspace , , ,
c
a rectorspace under inherited es bes , ,
es.
operations . Notice,
[83 Axiom I
&
·
I rivial subspace : Let 2
6=
1 .
, ,
W V
Improper subspace =
es bus = S be S .
·
:
, ,
·
Proper subspace : WE
Axiom 2 C
,
= =
C .
,
Propection A novemptysubseta
Note o is inherited
from V
,
so t is
only if associative .
1 w, 2W ,
V
,,, W
2 creW
,
VeeR ,
weW Axiom 3 ,
Let 2
, ==
0 .
05 105 ,
,
= 58 Jes =
.
Proof is
straightforward.
Axiom 4 ,
Let c
,
= 1
, 4
=
Proposition Let SEV ,
(V ,
b
,
%) is a observe , 2
,
5S
S
ventorspace S .
D The following
S
.
= St
=
-
5 =
y
are
equivalent.
Ssubspace of V
.is a Axiom 5 commutativity
I
25 eS c ,R , 5,eS , ,
is inherited from .
V
IS is closed under lines combos I
of 2 vectors ) .
3
VSi ,
Kin eS
, ,
citIR ,
s ,
es
, Span : Linear Independence Exercise Let A = Ex, . .
. 3c V
.
Suppose A is independent .
Definition Let unEV V
v vsp a Prove all subsets of A
.
....., ,
is a
. .
are
Let 2,
,
. .
.,
entR Then < 5 .
, ,
0 ... cUn linearly independent .
is called a liner combination of .
b If A were dependent ,
, In For a , ... .
subset W of V
, would a still hold ?
the Span(W) is the set of all
linear combinations of vectors in .
W
Definition . A basis for vectorspace
Terminology :
W is called a
spanning
V is a linearly independent
set of span(W). spanning set .
Lemma .
In a
vectorspace V the spar
, ,
of any set SSV is a Linear Transformations (CTSI)
subspace .
Definition. Let VW be a
ventorspace .
Definition .
Let VieV for ic[n] ·
If there A function 2: V-W is a
,
exist citR ,
1 :* 0 ,
such that linear transformationif for all
c
,
v, b .. -
bran = 0
,
then V , , . . In are , EV3 CelR ,
linerly dependent. If the only solution 1 ((n ov) L(u)v((v) =
to the above for all
· L(c) cL(v)
equation
=
is
then we say
3
ci 0 =
, , ... Un se
linearly independent. Lemma .
Let F : V -W be a function
Then the V
following an equivalent VCIR V-
, , ,
If is a linear transformation
f(c , , acf(v)
Theme c *) f(t)
2 =
c
N , ,
·
f) Ff(n) =
a Evi] an lim .
dep .
17 Jexo is
a solution to Ac = 5 .
Remark· I distributes over any finite sum
b .
Ev 3 , are independent of the only of rectors from V.
Ac=O is
5
solution to e = .
c .
JeV is in SpanEV3+ iff
= a solution to Ac = b .
Lemma .
Any collection of Kan rectors
in IR" is dependent .
, Example .
F : /" >
- R3 ; (5) +(3) To be in I/V W) , , efocy must be
a liner transformation
(ii) (5)
.
pf .
Let v .
=
82 = <R? coR
.
,
Observe ,
2)(i) (2) 2)(ii) =
S
-
(3(a)(3) Recall D
(For futur use) Definition . For an
(3)(3) 2(v )02(v)
.
-
=
,
nxn matrix M let [Mn , , u
=
,
Mne-man J
that is let in be the nth
,
row of .
M
Next consider
,
((c) =
((c()) (() = Then , define the determinant of M:
det : Mnxn- IR such that,
-
(3) (i) = det(M) det ( +2
det i
, Tn) =
,....
2
,
,
. . .
,x
.
Firti ,
. . .
, In) Getti =
, ...
Ti ...
Mn)
-
c(i) ix]
~
dete , .... Fi , ...i fi .
. .
.. en) -dette ... j.. ... In)
=
, : ...
Thus L is a liner transformation .
I
i *
]
4
dete , . . ., Ki , . . .
. [r) = K .
det[, . .
., en)
5
det(1) = 1 , where is an idembity
Proposition . Let V W
,
be
ventorspaces .
Let matrix
.
2)V W) ,
be the set of all lin. transformation
from V to W .
I(V W) forms ,
a
Vectorspace IR under
over
componentwise
addition.
Recall Week I worksheet problem 1 .
She can show I/V W) ,
is a vsp
by showing that I/V ,W) is a subspace
of F(S) .
Consider z : V-W .
W has Ow ,
since
W is a
vsp , define 7: >Wit Or .
zeI)V , W) ,
and thus I is non
empty -
Take ,R ,
f ze](V W)
, ,
.
Recall function addition is
(f j)(f)
+ =
f(x)oy(v)
