Assignment 7 Math 2270 Dylan Zwick Fall 2012 - University of Utah

[Pages:12]Math 2270 - Assignment 7

Dylan Zwick Fall 2012

Section 3.5 - 1, 2, 3, 20, 28 Section 3.6 - 1, 3, 5, 11, 24

1

3.5 - Independence, Basis, and Dimension

3.5.1 Show that v 1 , v 2 , v3 are independent but v 1 , v 2 , v 3 , v4 are dependent:

v'o0/

( VI *

) v=( 1 \0/

) v 3 =1 lj cI

v 4 = 3 (24\)

/

/0)

0 /

LQ

(3

liei7 (

Cl V t ? V +

-

1i

i1

/

V3

J1Y i4 j de,idr

40) / (el) (-1) ? ? (- I) I (0

501

-

t

-

q

flU( bfe

Vi / L/ 1 V3

2

q I ieqi e/9e4de,?1

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ca.

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ii

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cc

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II

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CD

CD

0

C')

CD

CD CD

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3.5.3 Prove that if a = 0 or d = 0 or f 0 (3 cases), the columns of U are

dependent:

O

V QC

7 a b \ de

U=(\O0 0 f/

(lni

15

/

I 17 0

0

dl

He O)

co

re

f

)

/oeJ

oJ

co)uninJ

+Jeii i'j

o)

4

3.5.20 Find a basis for the plane x 2y + 3z = 0 in R 3 . Then find a basis -- for the intersection if that plane with the y plane. Then find a basis for all vectors perpendicular to the plane.

1k e p

x

-

H

( (i - 3) / X\ y)

2

YI/

/

U1

I

0L

( ii

w

/

( yIl

iQ1)

C

he f/due U *J

7)

(iT

5

3.5.28 Find a basis for the space of all 2 by 3 matrices whose columns add to zero. Find a basis for the subspace whose rows also add to zero.

49

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/oo) /obo /

C)/

01/ (

/0 -h

6

3.6 - Dimension of the Four Subspaces

3.6.1 (a) If a 7 by 9 matrix has rank 5, what are the dimensions of the four subspaces? What is the sum of all four dimensions?

(b) If a 3 by 4 matrix has rank 3, whare are its column space and left nulispace?

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c+r I77

h) (/4

o j1j

I) a J-iM

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0

7

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