Quiz1 Fall 00.Ps
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Transcript of Quiz1 Fall 00.Ps
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8/18/2019 Quiz1 Fall 00.Ps
1/4
1 8 . 0 6 P r o f e s s o r S t r a n g Q u i z 1 O c t o b e r 2 , 2 0 0 0
Y o u r n a m e i s :
P l e a s e c i r c l e y o u r r e c i t a t i o n :
1 ) M 2 2 - 1 3 1 H o l m 2 - 1 8 1 3 - 3 6 6 5 t s h @ m a t h
2 ) M 2 2 - 1 3 2 D u m i t r i u 2 - 3 3 3 3 - 7 8 2 6 d u m i t r i u @ m a t h
3 ) M 3 2 - 1 3 1 H o l m 2 - 1 8 1 3 - 3 6 6 5 t s h @ m a t h
4 ) T 1 0 2 - 1 3 2 A r d i l a 2 - 3 3 3 3 - 7 8 2 6 f a r d i l a @ m a t h
5 ) T 1 0 2 - 1 3 1 C z y z 2 - 3 4 2 3 - 7 5 7 8 c z y z @ m a t h
6 ) T 1 1 2 - 1 3 1 B a u e r 2 - 2 2 9 3 - 1 5 8 9 b a u e r @ m a t h
7 ) T 1 1 2 - 1 3 2 A r d i l a 2 - 3 3 3 3 - 7 8 2 6 f a r d i l a @ m a t h
8 ) T 1 2 2 - 1 3 2 C z y z 2 - 3 4 2 3 - 7 5 7 8 c z y z @ m a t h
9 ) T 1 2 2 - 1 3 1 B a u e r 2 - 2 2 9 3 - 1 5 8 9 b a u e r @ m a t h
1 0 ) T 1 2 - 1 3 2 I n g e r m a n 2 - 3 7 2 3 - 4 3 4 4 i n g e r m a n @ m a t h
1 1 ) T 1 2 - 1 3 1 N a v e 2 - 2 5 1 3 - 4 0 9 7 n a v e @ m a t h
1 2 ) T 2 2 - 1 3 2 I n g e r m a n 2 - 3 7 2 3 - 4 3 4 4 i n g e r m a n @ m a t h
1 3 ) T 2 1 - 1 5 0 N a v e 2 - 2 5 1 3 - 4 0 9 7 n a v e @ m a t h
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8/18/2019 Quiz1 Fall 00.Ps
2/4
1 ( 3 0 p t s . ) S u p p o s e t h e m a t r i x A
h a s r e d u c e d r o w e c h e l o n f o r m R :
A =
2
6
6
6
4
1 2 1 b
2 a 1 8
( r o w 3 )
3
7
7
7
5
R =
2
6
6
6
4
1 2 0 3
0 0 1 2
0 0 0 0
3
7
7
7
5
( a ) W h a t c a n y o u s a y i m m e d i a t e l y a b o u t r o w 3 o f A ?
( b ) W h a t a r e t h e n u m b e r s a
a n d b ?
( c ) D e s c r i b e a l l s o l u t i o n s o f R x = 0 . C i r c l e t h e s p a c e s t h a t a r e t h e s a m e
f o r A
a s f o r R
: ( r o w s p a c e ) ( c o l u m n s p a c e ) ( n u l l s p a c e ) .
2
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8/18/2019 Quiz1 Fall 00.Ps
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2 ( 3 0 p t s . ) ( a ) F i n d t h e n u m b e r c
t h a t m a k e s t h i s m a t r i x s i n g u l a r ( n o t i n v e r t i b l e ) :
A =
2
6
6
6
4
1 2 3
1 5 6
2 6 c
3
7
7
7
5
( b ) I f c
= 2 0 w h a t a r e t h e c o l u m n s p a c e C ( A
) a n d t h e n u l l s p a c e N ( A
) ?
D e s c r i b e t h e m i n t h i s s p e c i c c a s e ( n o t j u s t r e p e a t t h e i r d e n i t i o n s ) .
A l s o d e s c r i b e C ( A
; 1
) a n d N ( A
; 1
) f o r t h e i n v e r s e m a t r i x !
( c ) W i t h c
= 2 0 f a c t o r t h e m a t r i x i n t o A =
L U ( l o w e r t r i a n g u l a r L
a n d
u p p e r t r i a n g u l a r U
) .
3
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8/18/2019 Quiz1 Fall 00.Ps
4/4
3 ( 4 0 p t s . ) S u p p o s e A
i s a n m
b y n
m a t r i x o f r a n k r .
( a ) I f A x = b
h a s a s o l u t i o n f o r e v e r y r i g h t s i d e b
, w h a t i s t h e c o l u m n
s p a c e o f A ?
( b ) I n p a r t ( a ) , w h a t a r e a l l e q u a t i o n s o r i n e q u a l i t i e s t h a t m u s t h o l d b e -
t w e e n t h e n u m b e r s m , n , a n d r .
( c ) G i v e a s p e c i c e x a m p l e o f a 3 b y 2 m a t r i x A
o f r a n k 1 w i t h r s t
r o w 2 5 ] . D e s c r i b e t h e c o l u m n s p a c e C ( A
) a n d t h e n u l l s p a c e N ( A )
c o m p l e t e l y .
( d ) S u p p o s e t h e r i g h t s i d e b
i s t h e s a m e a s t h e r s t c o l u m n i n y o u r e x a m p l e
( p a r t c ) . F i n d t h e c o m p l e t e s o l u t i o n t o A x = b .
4
