MATH 260 LINEAR ALGEBRA EXAM III Fall 2014

Transcription

1 MAH 60 LINEAR ALGEBRA EXAM III Fall 0 Instructions: the use of built-in functions of your calculator such as det( ) or RREF is permitted ) Consider the table and the vectors and matrices given below Fill in each row of the table completely; each row is worth up to two points You needn t show any work unless you wish to a) b) c) d) e) V R V R V R Given the vector space V and the vectors v v v n : Are the vectors linearly independent? Do they span the given vector space? 5 7 Yes No Yes No Yes No Yes No 9 Yes No Yes No V P x xx x x Yes No Yes No V P x x xx xx x x x Yes No Yes No Math 60 Exam Fall 0/Page of 6

2 ) Consider the table and the vectors and matrices given below Fill in each row of the table completely; each row is worth up to two points You needn t show any work unless you wish to a) b) V R V R c) V M d) e) Given the vector space V and the vectors v v v n : Do they span the given vector space? Are they usable as a basis for the given vector space? 0 Yes No Yes No 0 0 Yes No Yes No Yes No Yes No V P x xx x x Yes No Yes No V P x x xx xx x x x Yes No Yes No ) Find a basis for and the dimension of the following subspaces: a) x 0 V A A ; xy 0 y R a subspace of M V u u xx yyx y; xy R a subspace of b) R ) Can we use the vectors u x addition and scalar multiplication? u x and u x x as a basis for P with the usual 5) Determine whether the following statements are true or false If the statement is true justify why it is true If the statement is false explain why or provide a counter-example a) If V is a vector space and v v v V with n span v v vn V then v v v n are linearly independent b) If V is a vector space and v v v V with n span v v vn V then dimv n Math 60 Exam Fall 0/Page of 6

3 For questions 6-8 let A ) a) Find the rank of A b) A basis for the column space of A 7) a) Find the nullity of A and b) A basis for the null space of A 8) Consider the linear system Ax 0 b given by the augmented matrix below a) Find a solution for the homogeneous system and express it in vector form b) It turns out that a particular solution of the system is given below Use this along with a portion of your results from above to find a general solution of the system Ax b : xxp x h using the nonzero column vector b given in part a) he particular solution: x x x x x x Math 60 Exam Fall 0/Page of 6

4 9) For each part use the information given in the table below to find the dimension of the row space of N A C A the column space of A C A and the null space of A A (a) (b) (c) (d) (e) Size of A 6 5 Rank(A) dim a) C A dimc A dimn A b) C A dimc A dimn A dim dim c) C A dimc A dim N A d) C A dimc A dim N A dim dim e) C A dimc A dimn A 0) Consider two bases of R V and W a) [ points] Find the transition matrix from V to W 0 0 b) [ points] Find the coordinate vector of x c) [ points] If x W what is x? W relative to V x V Math 60 Exam Fall 0/Page of 6

5 ) Consider the inner product p q p x q x dx on P a) [ points] If px x and q x x b) [ points] Find p c) [5 points] Find d p q find p q ) a) [ points] Use Gram-Schmidt to produce an orthonormal basis for inner product aka the dot product) from the vectors 8 R (using the Euclidean v and v 8 6 b) [6 points] Using the graphs provided below draw the vectors from the basis orthogonal basis orthonormal basis u u on the right graph below Label everything carefully v v and the w w on the left graph below Label everything clearly Draw the Math 60 Exam Fall 0/Page 5 of 6

6 ) BONUS [0 points] You can earn two points for each row of the table below that is correctly filled out Assume A is an m n matrix; put a checkmark in the appropriate box inside each cell to indicate if the situation is possible or not For example if you were told that A was with rank(a) = then you would mark Yes only in the unique solution column because in that case it isn t possible for A to be inconsistent or dependent In each cell provide brief justification (no justification no credit) 5 Ax b No solution (inconsistent) One solution Infinite solutions A is with rank Yes No Yes No Yes No A is with rank as large as possible Yes No Yes No Yes No A is with rank as Yes No Yes No Yes No large as possible A is with rank Yes No Yes No Yes No A is with rank and b 0 Yes No Yes No Yes No ) BONUS [0 points] Find a basis for and the dimension of the following subspace: V px pxa0 axax a x px is odd a subspace of P 5) BONUS [0 Points] Determine whether the following statements are true or false If the statement is true justify why it is true If the statement is false explain why or provide a counter-example a) If the vectors uvw are linearly independent then uv are linearly independent as well b) If the vectors uv are linearly independent then u v u v are linearly independent as well 6) BONUS [0 Points] Find all matrices such that: a) he null space is the line x y 0 b) he null space is the line xy 0 Math 60 Exam Fall 0/Page 6 of 6