Corresponding Homogeneous System - Linear Algebra - Exam Solution, Exams of Linear Algebra

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MATHS348 - March 1, 2010 NAME: Key. Exam I - 50 Points - 50 minutes SECTION: In order to receive full credit, SHOW ALL YOUR WORK, Full credit will be given only if all reasoning and work is provided. When applicable, please enclose your final answers in boxes. 1. (10 Points} (a) True/False : Mark each statement as either true or false. i, Suppose that Ax = b, where A € R"*”, has no solutions. The corresponding homogeneous system, Ax = 0, has only the trivial, x = 0, sohition. . False ii, If A ¢ R™*" has a row of zeros then Ax = 0 always has infinitely-many solutions. False. See Hwa Pr mordix As iil. It is impossible for a vector to be in both the null-space a: column-space of a matrix. False [Se Hw#2 > LL iv. If the dimension of the colunth-space of Anyn is m then Ax =0 has only the trivial solution. Tr v. The system Ax = 0, where A € R°*4 has only the trivial solution. See Slide on False [eat $3 shone (b) Short Response : Provide a sfort justification of your conglusion. i, Suppose Vaxn is a matrix whose columns form a basis for R®, What can be said about the determinant of V? Columns of V ~»Y ~~ =D det ly) #0 $form, “a basis foe \R” fi, Suppose that A = 0 is an eigenvalue of A. What can be said about A~!? det (A~T) = det(a)= 0 => A’ docs not Bright on wl AR dR? O- KD R ida novkwial sola to Ana =? A DNE