Math 110 MidTerm Test March 5 2002 11 43 am At the request of the class a section of Math 110 will submit to a 50 min Closed Book Midterm Test during one of the three lecture periods Mon Wed Fri 1 2 pm in the week of 4 8 March 2002 This means that students must put away all books and papers and computing instruments before the test begins It will be presented on one sheet of paper containing questions and blank spaces for answers Each correct answer will earn one point each incorrect answer will lose one point Each answer left blank or scratched out will earn or lose nothing Therefore mere guesses make poor answers TOPICS for the Math 110 Midterm Test 4 8 Mar 2002 area volume higher dimensional content associativity of addition and multiplication basis bases change of basis codomain corange cokernel of linear operator column echelon form reduced column echelon form content like area and volume in Affine spaces commutativity of addition but not complementary projectors cross product of vectors in Euclidean 3 space determinants properties like det B C det B det C det BT det B dimension of a linear space distributivity of multiplication over addition domain of a linear operator dual spaces of linear functionals dyad rank one linear operator elementary row and column operations dilatators shears existence and non existence of solutions of linear equation systems fields of scalars hyperplanes equations of hyperplanes inverses of linear operators and matrices L 1 length of a vector Euclidean length linear spaces affine spaces Euclidean spaces linear functionals linear dependence and independence linear operators lines equations of lines parametric representation of a line norm of a vector Euclidean length null space or kernel of a linear operator orientation of area volume higher dimensional content parallel lines parallel hyper planes parallelepipeds permutations odd and even planes equations of planes parametric representation of a plane projector P P2 Prof W Kahan Page 1 5 This document was created with FrameMaker 4 0 4 Math 110 MidTerm Test March 5 2002 11 43 am range of a linear operator rank row rank column rank determinantal rank reflection in a hyper plane in a line in a point rotations in Euclidean 3 space row echelon form reduced row echelon form singular non invertible matrix span of subspace spanned by a set of vectors target space of a linear operator transpose of a matrix triangular matrix triangular factorization uniqueness and non uniqueness of solutions of linear equation systems vectors vector spaces volume higher dimensional content Relevant Readings these notes are posted on the class web page http www cs berkeley edu wkahan MathH110 2Dspaces pdf Cross pdf For this test you need not memorize triple vector product identities nor the formulas on pages 7 11 GEO pdf GEOS pdf but not pages 4 6 for this test RREF1 pdf TriFact pdf pts pdf but for this test skip the last paragraph on p 8 and what follows Prof W Kahan Page 2 5 Math 110 MidTerm Test March 5 2002 11 43 am This is a Closed Book Midterm Test for Math 110 Student s SURNAME ANSWERS GIVEN NAME Students must put away all books and papers and computing instruments before the test begins Its one sheet of paper contains questions and blank spaces for answers Each correct answer earns one point each incorrect answer loses one point Each answer left blank or scratched out earns or loses nothing Therefore mere guesses make poor answers Only answer blanks contents will be graded so the rest of the sheet can be used for scratch paper 1 Can the columns of a 3 by 4 matrix Sometimes Never be linearly independent CHOOSE ONE BY WRITING X IN A BOX Answer Never 2 A Tetrahedron is a figure with four vertices six edges and four triangular faces each face is opposite one vertex and bounded by three edges through the other three vertices The tetrahedron is called nondegenerate if no vertex lies in a plane containing the opposite face Given any two nondegenerate tetrahedra each with a vertex at the origin in a 3 dimensional space either can be mapped onto the other by an invertible Linear Operator always in infinitely many ways always in finitely many ways more than one always in just one way sometimes not always never CHOOSE ONE BY MARKING X IN A BOX Answer always in six ways corresponding to the six ways to send one tetrahedron s three edges emanating from the origin to the other s There would be infinitely many ways if the tetrahedrons were situated in a space of more than three dimensions can you see why Think of the edges emanating from the origin as basis vectors 3 Knowing only that matrices B C and D satisfy B C D I an identity matrix may we infer that C has an inverse and if so can it be expressed in terms solely of B and D Always and C 1 Sometimes C 1 if B and D are No sometimes C has no inverse CHOOSE THE TRUE STATEMENT S BY FILLING THE BOX ES WITH X AND FILL IN ANY BLANK AFTER YOUR CHOICE S Answer If B and D are square in which case they must be invertible too do you see Prof W Kahan Page 3 5 Math 110 MidTerm Test March 5 2002 11 43 am why then C 1 D B But otherwise C may have an inverse or it may not for example 1 o T T 1 o o H 1 o 1 but the middle matrix C has no inverse unless H has one 4 Linear operator L is representable by a 4 by 3 matrix The set of all solutions z of the equation L z o sweeps out a two dimensional subspace a plane through the origin o Two different vectors b L u and c L v are known to be nonzero What is the dimension of the range of L FILL IN A NUMBER Answer 1 Vectors b and c must be anti parallel because L has a two dimensional null space in a three dimensional domain and therefore the range of L must must be a subspace of dimension 3 2 1 within L s four dimensional target space 5 Complex numbers 1 form a 2 dimensional vector space over the real number field these vectors can be multiplied to produce other vectors in the same space and this multiplication is both commutative and associative The cross product u v of vectors in a 3dimensional Euclidean space is anti commutative v u u v and non associative only in special cases does u v w u v w Real 2 by 2 matrices constitute a 4 dimensional real vector space whose vectors can be multiplied to produce other vectors in the space its multiplication is associative but not generally commutative Is there a 3 dimensional vector space over the real field whose vectors can be multiplied to produce other vectors in the same space and whose multiplication is