lz8el i'l; i j- [l i; ir { L:,
Transcript of lz8el i'l; i j- [l i; ir { L:,
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No calculators or ang other deaices alloued,If ang quest'i,on i.s not clea,r, ask for clari,fi.cat'i,on.No cred.i,t ui,II be g'i.aen for i.llegi.ble solutions.If gou present d.i,fferent answers for the sa,rne problern,the uorst arlsurer ui,ll be graded.Show all your work. lBou your ansuters.l
MTH 4L5Exam LJuIy 21-, 201050 minutesSects: L.'J.-L.4, 2.L-2.6
1. 1tS points) Express matrix A as a sum of a symmetric matrix and a skew-symmetricmatrix, where
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A
r i r z 3l i lq Tl , iei ' l ; ? i j - [ l i ; i r " { L: ,b] I+l l .
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2. OS points) Determine which of the following functions ?, ,5 : 1R2 -- R2 is iinear:
r (1", \ : I r , ,1. sf [ , ' l ) : f , , -* ' , .1'\1" ' .1 )- lsrprt \Lz2l l Ls ' ' JIf a function is linear, give a proof; if a function is not iinear, show it with an example.
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lx l3'4ib ry
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lulr*obj X+ bx j j
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l], o"o matrix X solution(20 points) Given the matrices A :
of the matrix equation
A{x,4-d,k i e -? AB
YA-(>( * * f b
x {A- 6r\ . -8 D A-(,y, {*
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*f
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- * f t
f " - r -b iI
i i' a ' t ii '?i . - l
r.' J= 1l E
Lb t I)
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t
(20 points) Consider the matrices '
f r 2zfR: | 3 2 11,
L-3 2 7J
Verify whether the foliowing equations hold:
4.
FA
N(A) : .^/(B)?, l/(A') : l /(B')?, n(A) :,R(BX'
Justify your answers.
R(A') : E(B')?.
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2-e l l r o, Q. i -o l o to o I Lo o
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l la
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t*
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l ' Bit
Ll3 |
= f ,I :
Att
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iro'r>lulL0 0fr
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5. (tS points) Determine whether the following statement true or false:
"Given a matrix A € IF-'", it holds that tr (A'A) :0 if and oniy if A:0."
If the statement is true, prove it; if the statement is false, give an example showing it.
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6. (fS points) Find the LU-factorization of matrix A below and use it to find the solutionvector x of the linear svstem Ax: b, where
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s) Q' t = - | '+ E'3r3
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