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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 
11 "Math 2270-1" }}{PARA 257 "" 0 "" {TEXT 257 27 "Review, and Practic
e Exam 2" }}{PARA 258 "" 0 "" {TEXT -1 17 "November 8, 2000 " }}{PARA 
275 "" 0 "" {TEXT -1 0 "" }}{PARA 276 "" 0 "" {TEXT 260 18 "Review Inf
ormation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
227 "     The exam will cover chapters 4, 6 and sections 8.4, B.1.  In
 addition to being able to do the computations from these sections, yo
u should know key definitions,  the statements of the main  theorems, \+
and why they are true." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 262 10 "Chapter 4:" }}{PARA 0 "" 0 "" {TEXT -1 45 "     Vec
tor spaces  (definition and examples)" }}{PARA 0 "" 0 "" {TEXT -1 48 "
          subspaces (how to check for, examples)" }}{PARA 0 "" 0 "" 
{TEXT -1 43 "          linear combinations  (definition)" }}{PARA 0 "
" 0 "" {TEXT -1 72 "          linear independence/dependence  (definit
ion, how to check for)" }}{PARA 0 "" 0 "" {TEXT -1 55 "          span \+
(definition, how to find good basis for)" }}{PARA 0 "" 0 "" {TEXT -1 
142 "          basis  (definition, how to extract a basis from a spann
ing set, good bases, how to create a basis from a set which does not y
et span" }}{PARA 0 "" 0 "" {TEXT -1 34 "          dimension  (definiti
on, " }{TEXT 261 3 "why" }{TEXT -1 51 " does each basis have the same \+
number of elements?)" }}{PARA 0 "" 0 "" {TEXT -1 106 "          soluti
on space of homogeneous systems, i.e. matrix nullspace  (definition, h
ow to find basis of)" }}{PARA 0 "" 0 "" {TEXT -1 127 "          the fo
ur fundamental subspaces of a matrix (there were three until section 4
.9) (what are they, how do you find them)" }}{PARA 0 "" 0 "" {TEXT -1 
56 "          rank and  nullity (definitions, theorem about)" }}{PARA 
0 "" 0 "" {TEXT -1 112 "          coordinates and change of basis  (de
finition of coordinates, how to change bases for subspaces of R^n)" }}
{PARA 0 "" 0 "" {TEXT -1 88 "          orthogonal and orthonormal base
s  (definition, how to create via Gram-Schmidt)" }}{PARA 0 "" 0 "" 
{TEXT -1 89 "          projection onto subspaces of R^n  (via orthogon
al bases or not; Method 1 and 2)" }}{PARA 0 "" 0 "" {TEXT -1 9 "      \+
   " }}{PARA 0 "" 0 "" {TEXT 263 29 "8.4:  Method of least squares" }}
{PARA 0 "" 0 "" {TEXT -1 108 "          least squares solutions to inc
onsistent matrix equations  (geometric meaning, method of computing)" 
}}{PARA 0 "" 0 "" {TEXT -1 74 "          application of above method t
o linear or quadratic data fitting." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 264 37 "B.1 Inner Product spaces: not on exam" 
}}{PARA 0 "" 0 "" {TEXT -1 122 "         abstract dot products lead to
 ways of measuring distance, angle and projection in vector spaces oth
er than R^n.  " }}{PARA 0 "" 0 "" {TEXT -1 10 "          " }}{PARA 0 "
" 0 "" {TEXT 265 11 "Chapter 6: " }}{PARA 0 "" 0 "" {TEXT -1 55 "     \+
     linear transformations (definition, examples)" }}{PARA 0 "" 0 "" 
{TEXT -1 52 "          kernel and range (definition, how to find)" }}
{PARA 0 "" 0 "" {TEXT -1 81 "          matrix of a linear transformati
on (definition, how to find, how to use)" }}{PARA 0 "" 0 "" {TEXT -1 
75 "          effect of basis changes on the matrix of a linear transf
ormation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT 
259 13 "Practice Exam" }}{PARA 0 "" 0 "" {TEXT -1 227 "     This exam \+
is closed-book and closed-note.  You may not use a calculator which is
 capable of doing linear algebra computations.  In order to receive fu
ll or partial credit on any problem, you must show all of your work an
d " }{TEXT 258 25 "justify your conclusions." }{TEXT -1 121 " There ar
e 100 points possible, and the point values for each problem are indic
ated in the right-hand margin.  Good Luck!" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "1a)  Show that the following ma
trix equation has no solution" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%
'matrixG6#7%7$\"\"\"\"\"#7$!\"\"\"\"!7$F.F*F*-F&6#7$7#%#x1G7#%#x2GF*-F
&6#7%7#F+7#F.7#\"\"%" }}{PARA 265 "" 0 "" {TEXT -1 10 "(5 points)" }}
{PARA 0 "" 0 "" {TEXT -1 64 "1b)  Find the least-squares solution to t
he problem in part (a)." }}{PARA 266 "" 0 "" {TEXT -1 11 "(10 points)
" }}{PARA 0 "" 0 "" {TEXT -1 67 "2)  Let W be the span of the vectors \+
\{[1,-1,0],[2,0,1]\} in 3-space." }}{PARA 0 "" 0 "" {TEXT -1 47 "2a)  \+
Find an orthonormal basis for the plane W." }}{PARA 267 "" 0 "" {TEXT 
-1 11 "(10 points)" }}{PARA 0 "" 0 "" {TEXT -1 95 "2b)  Let v=[2,0,4].
  Find the projection of v onto the subspace W, using your answer from
 (2 a)" }}{PARA 268 "" 0 "" {TEXT -1 11 "(10 points)" }}{PARA 0 "" 0 "
" {TEXT -1 165 "2c)  If you did your computations correctly, your answ
er to part (2b)  should equal the matrix from number (1) multiplied by
 your least squares [x1,x2] to part (1b)," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&-%'matrixG6#7%7$\"\"\"\"\"#7$!\"\"\"\"!7$F-F)F)-F%6#7
$7#%#x1G7#%#x2GF)" }}{PARA 0 "" 0 "" {TEXT -1 12 "Explain why." }}
{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 10 "(5 poi
nts)" }}{PARA 0 "" 0 "" {TEXT -1 7 "3)  Let" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"LG6#-%'matrixG6#7$7#%\"xG7#%\"yG*&-F(6#7$7$\"\"\"
\"\"$7$F5F4F4F'F4" }}{PARA 0 "" 0 "" {TEXT -1 199 " be a matrix map fr
om R^2 to R^2.  Let S=T=\{[1,1],[-1,1]\} be a non-standard basis for R
^2. Find the matrix for L with respect to S and T.  You may do this pr
oblem either of the two ways we discussed." }}{PARA 270 "" 0 "" {TEXT 
-1 11 "(20 points)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 116 "4)  Let A be the four by five matrix below.  Also shown \+
are the reduced row and reduced column cecheolon forms of A." }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7'\"\"#\"\"\"!\"\"\"\"$
\"\"!7'F+F,F*F,F+7'F.F-!\"&\"\"&!\"#7'F+F.F.F+F+" }}{PARA 261 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "rref(A);  #reduced \+
row echelon form" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7'\"
\"\"\"\"!F)F(F(7'F)F(F)F)!\"%7'F)F)F(!\"\"!\"#7'F)F)F)F)F)" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 39 "rcef:=M->transpose(rref(transpose(M)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rcefGR6#%\"MG6\"6$%)operatorG%&arro
wGF(-%*transposeG6#-%%rrefG6#-F-6#9$F(F(F(" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "rcef(A);  #reduced column echelon form" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7'\"\"\"\"\"!F)F)F)7'F)F(F)F)F)7'F
(!\"#F)F)F)7'F)F)F(F)F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 103 "4a)  Find a ``good'' basis for the column space of \+
A, i.e. one whose vectors have lots of zero entries." }}{PARA 262 "" 
0 "" {TEXT -1 10 "(5 points)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 131 "4b)  Prove that the vectors you found in part \+
(a) actually are linearly independent, using the definition of linear \+
independence.  " }}{PARA 263 "" 0 "" {TEXT -1 12 "(10  points)" }}
{PARA 0 "" 0 "" {TEXT -1 41 "4c)  Find a basis for the nullspace of A.
" }}{PARA 264 "" 0 "" {TEXT -1 11 "(10 points)" }}{PARA 0 "" 0 "" 
{TEXT -1 142 "4d)  Find a basis for the orthogonal complement to the n
ullspace of A.  Verify that its basis elements are orthogonal to the n
ullspace basis. " }}{PARA 271 "" 0 "" {TEXT -1 10 "(5 points)" }}
{PARA 0 "" 0 "" {TEXT -1 71 "4e) Find a basis for the orthogonal compl
ement to the column space of A" }}{PARA 272 "" 0 "" {TEXT -1 10 "(5 po
ints)" }}{PARA 0 "" 0 "" {TEXT -1 98 "4f)  Verify the theorem which re
lates rank, nullity, and domain dimension, for our matrix A above." }}
{PARA 273 "" 0 "" {TEXT -1 10 "(5 points)" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{MARK "34 0" 80 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }