Math. 110 MidTerm Test March 5, 2002 11:43 amProf. W. Kahan Page 1/5 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 contentassociativity of addition and multiplicationbasis, bases, change of basiscodomain, corange, cokernel of linear operatorcolumn-echelon form, reduced column-echelon formcontent, like area and volume, in Affine spacescommutativity of addition but not ...complementary projectorscross-product of vectors in Euclidean 3-spacedeterminants’ properties like det(B·C) = det(B)·det(C) , det(B T ) = det(B) , …dimension of a linear spacedistributivity of multiplication over additiondomain of a linear operatordual spaces of linear functionalsdyad (rank-one linear operator)elementary row- and column-operations, dilatators, shears, ...existence and non-existence of solutions of linear equation-systemsfields of scalarshyperplanes, equations of hyperplanesinverses of linear operators and matrices: L -1 length of a vector, Euclidean lengthlinear spaces, affine spaces, Euclidean spaceslinear functionalslinear dependence and independencelinear operatorslines, equations of lines, parametric representation of a linenorm of a vector, Euclidean lengthnull-space or kernel of a linear operatororientation of area, volume, higher-dimensional contentparallel lines, parallel (hyper)planes, parallelepipedspermutations, odd and evenplanes, equations of planes, parametric representation of a planeprojector P = P 2 This document was created with FrameMaker404Math. 110 MidTerm Test March 5, 2002 11:43 amProf. W. Kahan Page 2/5 range of a linear operatorrank, row-rank, column-rank, determinantal rank, …reflection in a (hyper)plane, ... in a line, ... in a pointrotations in Euclidean 3-spacerow-echelon form, reduced row-echelon formsingular (non-invertible) matrixspan of (subspace spanned by) a set of vectorstarget-space of a linear operatortranspose of a matrixtriangular matrix, triangular factorizationuniqueness and non-uniqueness of solutions of linear equation-systemsvectors, vector spacesvolume, higher-dimensional content Relevant Readings: these notes are posted on the class web page http://www.cs,berkeley.edu/~wkahan/~MathH110 2Dspaces.pdfCross.pdf ( For this test you need not memorize triple-vector-product identitiesnor the formulas on pages 7 - 11.)GEO.pdfGEOS.pdf ( but not pages 4 - 6 for this test.)RREF1.pdfTriFact.pdfpts.pdf ( but for this test skip the last paragraph on p. 8 and what follows.)Math. 110 MidTerm Test March 5, 2002 11:43 amProf. W. Kahan Page 3/5 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 be linearly independent? [__] Sometimes. [__] Never. (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 αβγδεζηθκλµνMath. 110 MidTerm Test March 5,