solution6(1) - SECOND HOURLY PRACTICE VI Math 21b Fall 10...

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11/4/2010 SECOND HOURLY PRACTICE VIMath 21b, Fall 10Name:MWF10 Oliver KnillMWF11 Anand PatelStart by writing your name in the above box andcheck your section in the box to the left.Try to answer each question on the same page asthe question is asked. If needed, use the back or thenext empty page for work. If you need additionalpaper, write your name on it.Do not detach pages from this exam packet or un-staple the packet.Please write neatly and except for problems 1-3,give details.Answers which are illegible for thegrader can not be given credit.No notes, books, calculators, computers, or otherelectronic aids can be allowed.You have 90 minutes time to complete your work.1202103104105106107108109101010Total:1101
Problem 1) (20 points) True or False? No justifications are needed.1)TFA 4×4 matrix whose entries are all 4 has determinant 44.Solution:The determinant is zero because the matrix has identitcal rows.2)TFbracketleftBigg1000bracketrightBiggis an orthogonal matrix.Solution:It is an orthogonalprojection, not an orthogonaltransformation.3)TFEvery 2×2 matrix is diagonalizable over the complex numbers.Solution:The shear is a counter example. It can not be diagonalized even using complex numbers.There is an eigenvector to the eigenvalue 1 with geometric multiplicity 1.4)TFIfAis a projection onto a linear subspaceV, then im(A) =Vand ker(A) =V.Solution:The image is clearlyV. Everything perpendicular to the image is maped to the origin.One can see second part also using the formula ker(A) = (im(AT))and the fact thatA=ATfor a projection.5)TFIfAis a 3×3 matrix representing reflection about a line in3, thenAissymmetric.Solution:The projectionPonto a line is symmetric. Because the reflectionRsatisfiesR= 2PI,also the reflectionR= 2PIis symmetric.6)TFIfAis a matrix with orthonormal columns, thenvectorx=ATvectorbmust be a least-squares solution of the systemAvectorx=vectorb.2
7)TFIfArepresents the projection onto a line in2, thenAis similar tobracketleftBigg1000bracketrightBigg.

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