DOC PREVIEW
HARVARD MATH 21B - Linear Algebra with Applications

This preview shows page 1-2-3 out of 10 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 10 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 10 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 10 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 10 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

!!"#$%&'()*+(&,(-%'"./(012"3/.(4%$5(0661%#.$%&'78(9&'1%'"./(!:7$";7(.'<(-%'"./%=.$%&'!"#$%&'()*$$%+',-'.$$#'/)%$&01%)23#&$'4*55%)%+$*67'% 8 96$*#+&'6+4'&-&$%:&'#5' 4*55%)%+$*67'%896$* # + & ' # + % '%+0#9+$%)&'*+';)60$*0 % ' 6)%+#+7*+%6)<'=#)'%>6:;7%?'6',*#7 # @*&$':*@1$':#4%7'$1% ' ;#;976$*#+&'!!"2'6+4'#!"2'#5'$(#'*+$%)60$*+@'&;%0*%&'#56+*:67&',-'$1%'5#77#( *+@'+#+7*+%6)'&-&$%:A!B C 2!D 2$!! ! #$"$## ! #$" = − −    = − −  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− −  =− −    ?$16$'*&?'$1%'&#79$ * #+&'6)%'$1%'57#('7*+%&'#5'$1%' R %0$#)'5*%74A!B C 2!D 2! ! ## ! #− − − −  !L%%';6@%'DDI'#5'$1%'$%>$<2T%'0#974'9&%'6 ' 0 # : ;9$%)'$#'@%+%)6$ % ' $ 1 * & ' R%0$#)'5*%74'!#)'6'0#))%&;#+4*+@'4*)%0$*#+'5*%742?',9$'*$'$9)+&#9$'$16$'%R%+'(*$1 # 9$'$1%'6*4'#5'6'0#:;9$ % )'*$'*&'+#$'16)4'$#'6+67-U%'$1%'7#+@M$%):',%16R*#)'#5'$1%'&-&$%:<E#'560*7*$6$%'$ 1 * &'4*&09&&*#+?'7%$'9&'()*$% '1-!!?'#2'V'!!B'M'C!'M'#2'6+4'0'!!?'#2'V'#!D'M'!'M'#2<'.9)'$6&S'*&'$#4)6('6')#9@1'&S%$01'#5'$1%'R%0$#)'5*%74! ? 2 !B C 2! ? 2 !D 21 ! # ! ! #0 ! # # ! #− −   =− −      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   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`%&06;%a'5)#:')%@*#+'JJJ'&*+0%'$1%'R%0$#) & ' 67#+@'$1%',#9+46)*%&;#*+$'`$1%'#$1%)'( 6-<a'E1%'$)6^%0$#)-'(*77'6;;)#601'$1%'%89*7*,)*9:';#*+$'!C?C2<\'&*:*76)')%6& # +*+@'&1#(&'$16$'6'$)6^%0$#)-'&$6)$*+@'*+')%@*#+'JN'(*77'6;;)#601'$1%'%89*7*,)*9:'!C?C2<\'$)6^%0$#)-'&$6)$*+@'*+')%@*#+'J'16&'$1)%%'`#;$*#+&aA'J$'06+'6;;)#601'$1%'%89*7*,)*9: ';#*+$'!C?C2'(1*7%)%:6*+*+@'*+') % @*#+'J'6$'677'$*:%&?'#)'*$'06 + '`0)#&&'#R%)a'*+$#')%@*#+&'JJJ'#)'JN<'E1%'5*+ 6 7'#9$0#:%'(*7767(6 -&',%'$1%'&6:%A'$1%'$)6^%0$#)-'(*77'6;;)#601' ! C ? C2<\'$)6^%0$#)-'&$6)$*+@'*+')%@*#+'JJ'16&'$1%'$1)%%'#;$*#+&'^9&$'4*&09&&% 4 '!5#)')%@*#+'J2?',9$',%&*4% & '$16$'*$:6-'&%%:'0#+0%*R6,7%'$16$'6'$)6^%0$#)-'0#974'`:%)@%a'(*$1'$1%'!M6>*&'#)'$1%'#M6>*& ?' 6 ; ; )#601*+@'$1%%89*7*,)*9:'!I?W2'6+4'!W ? D 2 ? ')%&;%0$*R%7-<'"#$%?'1#(%R%)?'$16$'$1%)%'*&'67) % 6 4 -'6'!&$)6*@1$M7*+%2'$)6^%0$#)-6;;)#601*+@'!I?W2'5)#:'$1%')*@1$<'/9$'$)6^%0$#)*%&'06++#$':%)@%'&*+0%'$1%'$)6^%0$#)-'5#)'6'@*R%+'*+*$ * 6 7R679%'*&'9+*89%?'5# ) ' ; # & * $ * R %'6+4'+%@6$*R%'"-!$1*+S '6,#9$'*$b2<_%$'9&'&9::6)*U%A'6&'7#+@'6&'$1%)%'6)%'&#:%'6+*:67&'5)#:'%6 01'&;%0*%&';)%&%+$'*+*$*677-'!$16$'*&?'!!'6+4#!'6)%',#$1';#&*$*R%2?'$1%+'$1%'&-&$%:'(*77'%R%+$9677-'6;;)#601'$1%'%89*7*,)*9:'&$6$%'!C?C2<'J5'!!'≠'W'6+4#!'V'W?'$1%+'$1%'&-&$%:'(*77' 6 ; ; ) #601'!I?W2Y'*5'!!'V'W'6+4'#!'≠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 =    =  (*$1'6+'%89 * 7*,)*9:'&#79$*#+'!&?'52?'$16$'*&?'1'!&?'52'V'0'!&?'52'V'W<'J+':97$*R6)*6,7%'0670979& ? '-#9'7%6) +%4$16$'$1%'7*+%6)'6;;)#>*:6$*#+' # 5 ' 6 ' 59+0$*#+'1'!!?'#2'+%6)'6';#*+$'!&?'5 2 ' * & '@*R%+',-! ? 2 ! ? 2 ! ? 2 ! 2 ! ? 2 ! 21 11 ! # 1 & 5 & 5 ! & & 5


View Full Document

HARVARD MATH 21B - Linear Algebra with Applications

Documents in this Course
Review II

Review II

84 pages

math21b

math21b

27 pages

Syllabus

Syllabus

12 pages

Basis

Basis

2 pages

Basis

Basis

2 pages

Load more
Download Linear Algebra with Applications
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Linear Algebra with Applications and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Linear Algebra with Applications 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?