solution4 - FIRST HOURLY PRACTICE IV Math 21b Fall 10 Name...

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10/7/2010 FIRST HOURLY PRACTICE IVMath 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) TF questions (20 points) No justifications needed
There is a matrixAand a vectorvectorbsuch thatAvectorx=vectorbhas no solution andker(A) ={vector0}.
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4)TFIfAis a 3×3 matrix which is a rotation around a line in3then thecolumns ofAform a basis of3.Solution:The map is invertible and therefore, the columns form a basis.5)TFThe rank of a diagonal matrixAequals the number of non-zero entries inA.Solution:Every nonzero diagonal element will produce a leading 1.6)TFRow reduction produces a diagonal matrix which has leading 1 at the placeswhere the original entries were not zero.2
7)TFA=7/139/133
The composition of a shear and a rotation in the plane is invertible.Solution:Each of these transformations are invertible. So is their composition.9)TFThe linear space of cubic polynomialsax3+bx2+cx+dhas dimension 3.Solution:x3, x2, x,1 form a basis. The dimension is 4.10)TF

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