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# Linear Algebra MAT 022A

UCD

GPA 3.88

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This 18 page Class Notes was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Class Notes belongs to MAT 022A at University of California - Davis taught by Staff in Fall. Since its upload, it has received 75 views. For similar materials see /class/187425/mat-022a-university-of-california-davis in Mathematics (M) at University of California - Davis.

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Date Created: 09/08/15

Name Student ID Midterm MAT 022A 1 71108 You have 100 minutes You may only use a pencil or pen and the scrap paper that I provide No calculators7 notes or books You must show your work to receive full credit 1 Solve the linear system respresented by the following augmented matrix 10 points 72 2 2 0 3 1 3 16 4 2 4 22 1 2 Find all solutions to 1 2 3 x2 0 5 points 0 l 1 1 3 3 Calculate the following or7 if impossible7 say why 10 points IpbH CHM C7300 4 Solve 5 points 3 l 3 1 4 16 8 4 2 4 2 5 22 ll 2 2 2 3 6 0 0 7 3 7 4 5 Find the matrix transformation which given the vector f outputs the vector y 2 5 points 6 Show that if a matrix has an inverse the inverse is unique 5 points 7 For an n gtlt 71 matrix A de ne detA 5 points 8 Find the determinants of the following rnatrices 10 points 123 a432 023 waw DgtJgtCTJgtJgt 00wa HOHgtCTJ Hqgtco DgtJgtOOO HOOO HOHgtO 9 For A calj a 2 gtlt 2 nonsingular matrix7 nd A 1 in terms of an 112 121 and 122 5 points 10 Find the adjoint matrix and inverse of 10 points OHH woo coco 11 ls the function fx g 1 2 that fx y How do you know 5 points onto R2 ie for each y in R2 is there an x in R2 such 12 How many solutions does the following system have How do you know 5 points 1 0 71 1 0 2 1 72 2 0 3 0 73 3 0 Name Student ID Midterm MAT 022A Surnrner Session ll 82208 You have 60 minutes You may only use a pencil or pen and scrap paper No calculators7 notes or books Computations Do as much work as you feel necessary to compute the following quantities 1 Compute the following if de ned7 or state otherwise 2points each 11 111111 al11l1111l 11 0001234 000 497173 b000 571047 000 38122 2 a Let A 1 i J and compute A2 7 3A if de ned or state otherwise3 points 0 b LetB0 8 8 J and compute B3 B if de ned or state otherwise 3 points 3 Compute the following if de ned or state otherwise a 2 points det1 1 1 1 1 b 3 points det 1 1 1 1 1 1 1 1 1 0 1 0 1 2 2 3 4 C 5 pomts det 5 0 0 1 71 0 0 0 4 Given 3 gtlt 3 matrices ABC so that detA 3detB detC 77 Compute 2 points each a det3A 1 b detATBO 1 C detCA 1B2 5 Given that L R3 a R2 is a linear transformation such that 1 1 0 1 0 2 L 0 1L 3 73M 0 Ol 0 0 7 1 Compute the following 2 points each a L OHO 6 Determine the standard matrix which represents each of the following linear transforma tions 2 points each a LR2 R2de nedbyL b LiRZ szhereweknowLgi andLlt3ig 7 Let u 17 27 71 72 and V 7137 713 and let 6 be the angle between the vectors uV Now 2 points each a Find Hull b Find 9 8 Give a short answer to each of the following questions You do not have to explain 2 points each a ls the transformation Ly x27y2 linear b ls the transformation Lxy7 x 7 y 2x 12 linear c True or False For any n gtlt n matrices A and B we have A BA B A2 2AB B2 d True or False lf detA0 then A must be the zero matrix e True or False lf AB are nonsingular matrices then AB 1 B lA l Worked Problems You must show your work to receive credit 9 Solve the linear system respresented by the following augmented matrix 7 points 72 2 2 0 3 1 3 16 4 2 4 22 10 Determine whether the following matrices are singular or nonsingular If nonsingular nd the inverse 4 points each 7123 a 3112 7110 1712 b 0 13 124 11 Determine whether or not 1 is a linear combination of 1 Spoints 12 Find three vectors um w so that u 1 u w but so that also 1 31 w5 points 13 Determine if 17 71 0 is in the range of the linear transformation L R3 a R3 given by Lz17 27374 1 372 x3731 2x2 2 8 points 14 Let x 17 2 7x be a nonzero n Vector in R and de ne 1 117 HXH Show that u is a unit vector 7 points Narne Student ID Final MAT 022A 1 8108 You have 100 minutes You may only use a pencil or pen and the scrap paper that 1 provide No calculators7 notes or books You must show your work to receive full credit 1 Let u and V be orthogonal vectors such that 2 and 1 Find H2u73v Your work must show your answer is true for any u and V that satisfy the above properties 10 points 1 0 2 Find an orthonormal basis for the subspace of R3 spanned by 1 7 1 10 points 0 2 3 Let N be the set of all 3 gtlt 3 nonsingular matrices with real entries Prove or disprove that N is a real vector space under the operations of matrix addition and scalar multiplication 10 points 4 Let Pn denote the set of all polynomials of degree at most 71 with real coef cients ie poly nomials of the form a0a1tagt2ant for 104117 an E R and let S t21tt34 15 points a ls the set S linearly independent in P3 b ls the set S a basis for P3 c What is d mPn7 no work necessary 1 2 4 8 i 5 LetA 1 1 2 4 15 pomts a Find a basis for the null space of A b Find a basis for the row space of A 0 Find a basis for the column space of A d What is the nullity of A e What is the rank of A 6 If nullityA 2 for some 6 gtlt 8 matrix A7 what is the rank of A 5 points 1 0 i2 0 1 0 0 0 7 S i 0 1 7 0 0 and T i 0 0 7 0 1 are ordered bases for the set of all 2 gtlt 2 diagonal matrices with real entries Find the transition matrix PSHT from the T basis to the S basis 10 points

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