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Introduction to Linear Algebra

by: Braeden Lind

Introduction to Linear Algebra MA 405

Marketplace > North Carolina State University > Mathematics (M) > MA 405 > Introduction to Linear Algebra
Braeden Lind
GPA 3.93

R. Ramsay

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R. Ramsay
Class Notes
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This 7 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 405 at North Carolina State University taught by R. Ramsay in Fall. Since its upload, it has received 22 views. For similar materials see /class/223682/ma-405-north-carolina-state-university in Mathematics (M) at North Carolina State University.

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Date Created: 10/15/15
Chapter 2 Review Questions Evaluate Let A be 3 gtlt 3 and suppose that lAl 2 Compute a l3Al 1 1 2 71 O 1 O 3 71 2 73 4 39 5 O 72 b l3A ll 013A 11 Find all values of a for which the matrix a2 O 3 5 a 2 3 0 1 is singular Answer each of the following as true or false and give brief explanations a b c d e f g detAAT detAZ detiA idetA IfAT A l then detA 1 IfdetA 0 then A 0 If detA 7 then Ax O has only the trivial solution IfA4 I then detA 1 IfA2 A and A 3E I then detA O Chapter 1 Review Questions xlx2 1372x43 Find all solutions to the linear system 2x1 x2 313 214 5 712 136x4 3 Find all values of a for which the linear system has a no solution b a unique solution and c infinitely many solutions 1 z 4 2x y 32 5 73xi3ya2 75aza78 1 2 71 If possible find the inverse of the following matrix 0 1 1 1 0 71 71 2 If A i2 2 find all values of A for wh1ch the homogeneous system A12 7 Ax O has a nontrivial solution 1 3 0 2 1 1 a IfA 1 0 1 1 and B 1 0 0 2 compute AB 1 1 1 4 1 1 1 1 0 2 2 b Solve Ax b for x if A 1 2 1 3 and b 1 4 2 5 3 Find the LU factorization of the coef cient matrix of the linear system Ax b Solve the system by using a forward substitution followed by a back substitution 2 2 71 3 A is 711 51 714 4 13 i7 75 Answer each of the following as true or false Explain your answers a IfA and B are n gtlt n matrices then A BA B A2 2AB BZ b If 111 and 112 are solutions to the linear system Ax b then w iul u2 is also a solution to Ax b c If A is a nonsingular matrix then the homogeneous system Ax O has a nontrivial solution d A homogeneous system of three equations in four unknowns has a nontrivial solution e If A B and C are n gtlt n nonsingular matrices then ABij1 C lB Chapter 6 Review Questions Exercise 11 page 311 Exercise 15 page 311 Find the characteristic polynomial eigenvalues and eigenvectors of A i 3 Find a basis for the eigenspace of each eigenvalue of A 0 0 CDHH 0030 000 0 i 0 Is A defective El 3 i2 1 Let A O 2 O If possible Find a nonsingular matrix S such that S lAS is diagonal 0 0 0 OOOJN ONOO 71 and Let A be a 2 gtlt 2 matrix whose eigenvalues are 3 and 4 and associated eigenvectors are 1 1 respectively W1thout computatlon nd matrix D that is s1m11ar to A and nons1ngular matrix S suchthat S lAS D For each of the following matrices determine if the matrix is diagonalizable a 0 0 b 0 0 2 0 Chapter 4 Review Questions Let L R2 gt R2 be de ned by re ection through the line ac y 0 a If x nd Draw a picture and use symmetry b Using part a verify that L is a linear transformation c Find the matrix representing L with respect to the standard basis on R2 Let L P3 gt R2 be the linear transformation de ned by Labaccac2 ac b a c a Find the matrix A which represents L with respect to the ordered basis 17 ac 2 on P3 and the standard basis on R2 b Find a basis for the kemel of L c If S is the subspace of P3 consisting of all polynomials of the form pac a 0002 nd a basis for the image of S LS The linear transformation L R2 gt R2 is represented by the matrix 0 with respect to the basis i 1 Find the matrix which represents L with respect to the standard basis Answers to Chapter 6 Review Questions pA A2 8A 12 A 2A 6 so the eigenvalues are A 72 76 The eigenspace of A is N A 7 AI and the rank plus the nullity of A 7 AI is n so the dimension ofthe eigenspace ofA is n 7 k P 1 A 6 3 8 7 A72 7 A eigenvalues 1 3 72 corresponding eigenvectors pA 2 i A247 A3 i A 76 A i 22A2 7 7A6 A i 22A i 1A 76 131 111 The eigenspace of A1 1 has basis L OJ the eigenspace of A2 6 has basis OJ and 0 0 101 0 the eigenspace of A3 A4 2 has basis 0 A is defective because an eigenvalue of 111 multiplicity two has a one dimensional eigenspace 1 2 1 Eigenvectors of eigenvalues 3 20 respectively form the matrix S 0 1 0 such that 0 0 73 300 S lAS 0 2 0 000 30 712 D0 4l s 1 1i 73 The following matrices have eigenvalues 1 2 2 The eigenspace of A 1 has basis 1 k 0 2 3 0 1 0 a A O 1 O Diagonalizable since NA72I hasbasis O O 0 0 2 0 1 J 2 3 1 f 1 b B 0 1 0 Not diagonalizable since NB72I has basislt 0 0 0 2 l 0 J Answers to Chapter 3 Review Questions I x 7 y 7 a x 1 7 a y A7fa11ss1ncea lt ygtia xa ybma yia x39 y y Mm A8 fails since 19 y x a IfMNareinVthenAMN AMANMANA MNA7 so M N is in V Also aMA aMA aAM AaM so aM is in V Thus V is a subspace ofRZXZ b Suppose that M 01 b is in V Then AM MA ie d ago bgd Z Thenc0andabdsoM bgd Z JerB 2ThusVhasdimension2andbasis3 a Linearlyindependent since 01uv02vw03wuO gt 1 0 1 01 0 0103u0102V0203w0 gt 1 1 0 Oz O gt 0 1 1 03 0 0102030 b Linearlydependent since uv7vwwz7zu0 x 7772 71 71 a xyz0xiyiz y y y 1 2 0 Basis z z 0 1 71 711 forVislt 1 7 O k 0 1 J b Extend the basis in part a to a basis for R3 by adding any vector in R3 which is not in V For example add e1 2 0 1 Let V1 V23 V3 V4 71 7 0 7 2 Thus V1 v2 V3 is abasis and V4 2v1 7 V2 3V3 OOOH OOHO OHOO 1 0 0 a The transformation matrix from basis S to basis 12x 1 is U 0 1 1 and 1 i2 3 1 0 0 8 1 0 0 8 8 O 1 174a0 1 072soLpS 2ie 1 i2 3 6 0 0 1 i2 72 191 8x2 1 7 21 7 2 i 2cc 3 b The transformation matrix from the S basis to the T basis is V 1 U where 2 1 0 the transformation matrix from basis T to basis 2 1 1 is V 1 0 1 and 0 3 0 2101 00 100 7 WU 1 0 10 1 1 0 1 0 i 1 V lU 0301723 001 g c Find the coordinate vector of p with respect to the T basis using parts a and b 3 MT V 1Ups 2 ie px 32x2 x 212 3 7 x 1 2 2 1 Let A 3 6 5 0 Reduce A to rref Find the rank ofA and find bases for the 1 2 1 2 rowspace the column space and the nullspace of A 1 2 2 71 1 2 3 6 5 0 gt O O 1 73 The rankA 2 A basis for the row space ofA is 1 2 1 2 0 0 0 0 r 1 2 1205 00173 Abasis for the column space ofA is 4 3 5 Abasis for l 1 1 J 1 the nullspace of A is l I l OOHM HWOO


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