solved algebra exam

8/12/2019 solved algebra exam

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30 pts.) No justification is required. No partial credit.

IPart I IWhich of the following statements are true? ut a T) before thecorrect ones and an F) before the wrong ones.I ) f in an n x n matrix such that I vl = Ivl for every vE lRn , then for

any bE IRn the equation ATX = bhas a unique solution.

T ) f A commutes with B then AT commutes with BT

T f 0 is an eigenvalue of A then rank A) < nF= ) f A is an n x n matrix with fewer than n distinct eigenvalues, then A

is not diagonalizable.

t= ) f A is an orthogonal matrix, then there exists an orthonormal eigen-basis for A

2------- /


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I artII I In this part we suppose A is a 3 x 3 matrix with eigenvalues 1 2and 3 Which of the following statements must be true? Put a T) beforethe correct ones and an F) before the wrong ones.1 ) There is some i E ]R3 so that the equation A - 3I x = v has no

solution.

F ) A is positive definite.\ ' ) A is invertible.

( ' \ ) A must be diagonalizable.

\ ) A can not be an orthogonal matrix.

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2 30 pts.) No justification is required. No partial credit.IPart I IFill in the blanks.

(a) The inverse of the matrix G ) is - ~ - - - I ; ~ > - - - = 3 : 1 - ~ + ) - - - - - -(b) f A is a 5 x 5 matrix with det(A) = 3, then rank(A) = =

1 1(c) The volume of the parallelogram defined by the vectors 1 and 2

2 2 ss __(d) f A is a 3 x 3 real matrix, and we know that 1 and 2 3i are two

eigenvalues of A then the trace of A is

(e) The dimension of the space of all matrices A that satisfies the equationG ~ ) ~ ~ A i S - = 2 _

4

r),.


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IPart II IForeachofthe followingproblem,writedownamatrix satisfyingthe givencondition.(f) A 2x 2non-diagonalmatrix with eigenvalues 1, 2.

(g) Asquarematrix that isnot diagonalizable. f \ 3 \ 0t; ( () D o 0 2.

(h) A 4x 5matrix with rank 3.r 2. , o

( 2 1- l7 I( 2 1) o).. U 2.. 2.

(i) An orthogonalmatrix allofwhoseentiresarenonzero..12 i -L.2l .. . .)..N -f 1 z..r42 -- 1 J~ { l l). 2.- rz.. i .2.. -

1j) A3x 3matrix with v= 2 as oneofits eigenvector.

e J

32 -( 0

lo 3 -2

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3 (15 pts.) Makeup your own problem: Give an exampleofa matrix A all ofwhoseentries are nonzero, and a nonzero vector b so that the solutions to theequation i=bformalinein~ 3 . Then findallsolutionsto yourequation.

A C C Y d t ~ -(;0 t I ~ r J ~ I we knew ~ f,cJ , _ 1:: c{r , f i \ i(j) A ) y xJ MchlX -@ R < . ~ F ( ~ ) ~ a ~ 3-t::2 WiJ 1;.

Rr e x . ~ ~ J we ~ ~ fJRh I\f F(/t t -b kR R ~ r A (t):: 0 O ')\0 I I , (

Vsi1 )11J &+)/ Cf trJ Jd sacA CM A IPk,e - e n 6 ~ hv 12e/D.; ,0 ~ f l ~ (0' r' , ' J ~ (1 ;:)f)~ w A:: r) t ~ ( ; )2> 3 )

d . : 6 ~ -6u A ~ ~ t {5 X; :: I) ( l /--tX; -:= t

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4 20pts.) LetB={1 x ,1 x 2 ,x x2 }(a) Showthat B is abasisofP2'(b) LetPo(x) = 1 x x2. Compute [PO(X)]B.(c) LetT :P2 ---+ P2 be the lineartransformationT p x)) = p( x) - p'x).

Writedown the matrixA ofthetransformationT withrespect to thebasis B.

(d) IsT an isomorphism?CCl), If o ~ Cl +X)f (t(r1;ft) - C; x-) Ci L) i ~ d C J ) X t CL f-4)X L

( r t ~ =0 I C -t ($-= c:. e2..f ( ..::-0 =9 C, f4.T- ~ ::0 ffc,f( +- CIt qt L J=0-:0(,::(Gf(1..-tC;) - cc;.f-Cj)av / -CL 0 / C3.:::0.

So 13, /;" as;} of f . [ d ~ ? . o ~ . ~ t ICb) +fi t X L ~ i ff-Xi-HXtf-V+XL)::::i) 11\ t-;\;Lk = ~(C) T ,+x)::: +:>\-/ ; t ~ f X H 6 1 - X ~ - f X l JT ( H X ~ : : : /ft-- 2)\ =.2Uf) tj - C+X) - (X-fXt)

T x-t > \ 2 ) ~ X t ~ ~ -2K i [ xtt- )f f-X l JU1XJ]1


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5 10 pts.) Calculate the determin nt of the m trixo 1 1 1 110111

A= 1 1 0 1 11 1 1011 1 1 1 0

[No partial credit!], II , 1 4- f I II -I 0 ) 0 0 - o \ vo f f i { A ~ o f g j \) \) t) o -i ~ C & J : Q I 0 i) ::: 1-- -{) --f) ( ~ ) ( V-J 0 0 ---/ D 0 0 0 -( D-0 ) 0 -- o.) D () f

( ( I IJ I , f( , ( , I /~ ( ~ d . ~ ~ 1 I I ( / I I I I I

=l ~ f l W } A f - / 1/.-=) dP = f -f) -VC- {) -J ):;:9-

10

c


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~ J \ j" . To I

: \S-I' )

1220 pts. Let A = 3 4 0

5(a) ind all eigenvalues of A(b) ind a basis for each eigenspace of A(c) Write down an invertible matrix P and a diagonal matrix D so that

P 1AP = D.

~ - ; l ) [ t l - ; \ ) H ) { ] S - ~ ) (A -SA -2)

iA0 !L _ J) D ? ~ I Z PA) .' ( 0 22011. l-tr ':2... Z~ J l J L-It- 0- 0 2 1 Jp 0- . ~ Q::f) D- Z0 { ~

b0

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20 pts.) Consider the quadractic form

(a) Write down the matrix A for q

(b) Find an orthogonal matrix Q nd a diagonal m trix D so th t =QTDQ(c) Diagonalize q(d) Is q positive definite?

t ) ~ I(c). 1;.- exX :; ff X - ~ )i(X,-lrf

J(X 14hXt f - ~ .

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solved algebra exam

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