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# Elem Linear Algebra MATH 341

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This 7 page Class Notes was uploaded by Henderson Lind II on Tuesday September 8, 2015. The Class Notes belongs to MATH 341 at University of Oregon taught by Staff in Fall. Since its upload, it has received 27 views. For similar materials see /class/187199/math-341-university-of-oregon in Mathematics (M) at University of Oregon.

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Date Created: 09/08/15

Math 341 H F 9 9 03 5 00 Practice Midterm Fall 2006 For each of the following matrices state the next row operation in the standard row reduction algorithm and then perform that row operation 1 1 1 1 1 0 elf 11 b 0 0 72 ml j ml 3 j e 0 0 0 0 2 1 2 0 1 2 1 0 Letu 0 V 1 andw 2 Find the set of all points in R3 that are contained in both the plane spanuv and the line spanw 1 73 0 4 Consider the matrix A 0 0 2 71 How many solutions are there to the 0 75 homogeneous equation Ax 0 Write the solution set as a parametric vector equation a Parameterize all solutions of the matrix equation Ai 6 where A 1 b With A as before parameterize all solutions of the equation Ai 2 3 For each of the following matrices indicate rst whether its columns are linearly inde pendent or linearly dependent and second whether it is invertible 1 1 2 74 1 0 2 1 71 2 1 1 2 a 0 0 g b 71 72 g c 72 0 0 g d 0 72 1 g e 0 2 1 3 2 3 0 71 0 0 0 0 3 De ne the terms Row Echelon Form and Reduced Row Echelon Form If you cannot give a precise de nition give several examples of each to receive partial credit Give an example of a set of vectors V1 Vn that are linearly independent and a linear transformation T such that the images TV1 TVn are linearly depen dent Hint Work backwards starting with a set of linearly dependent vectors and then construct T as a matrix transformation Find the matrix which represents the linear transformation from R2 to R2 which rst re ects across the line y then rotates 45 degrees clockwise then re ects across the x axis H O 1 71 0 1 9 Consider the matrix A 3 72 0 2 7 and let T R4 a R3 be the linear transfor 1 72 3 0 H D Find the inverse ofthe matrix A Prove that ifthe vectors 2 rnation given by TX AX a ls T onto b ls T one to one Please explain your answers and show your work OOH OMH 2 1 7 and use it to solve the equations Ai T 3 fora OOH a 1 andb 2 3 Prove that if T is a linear transformation then T6 1772 are linearly dependent7 then so are the vectors 2 2 7372 Math 341 Practice Midterm Solutions Fall 2006 1 For each of the following matrices state the nept row operation in the standard row reduction algorithm and then perform that row operation if a E E 1 at i 133 il E E a Add 712 times row 1 to row 2 b Exchange rows 2 and 3 c Scale row 2 by 71 d Add 4 times row 2 to row 1 e Add 72 times row 1 to row 3 2 1 0 2 Let u 0 V 71 and W 2 Find the set of all points in R3 that are 1 1 71 contained in both the plane spanu7 V and the line spanW The intersection of plane and a line is either empty ie7 they dont intersect at all7 a single point7 or a line ie7 the whole line is contained inside the plane First7 we observe that the origin 0 is always contained in the span of any set of vectors7 so our plane and line must intersect there Now7 we must determine if the entire line is contained in the plane7 or if 0 is their only intersection This is equivalent to asking whether or not W is contained in spanu7 V In other words7 we want to determine whether the equation aubv W has a solution Writing the corresponding augmented matrix and reducing it to row echelon form 2 1 0 2 1 0 2 1 0 0 71 2 a 0 71 2 a 0 71 2 1 1 71 0 12 71 0 0 0 Since we did not obtain a contradiction7 there must be at least one solution7 and so W is a linear combination of u and V Thus W7 and in fact the whole line spanW7 is contained inside the plane spanu7 V Alternatively7 we can solve the equation au bv CW for the parameters a7 bc Now we are dealing with a homogeneous equation7 and the reduced row echelon form is 2 1 0 2 1 0 1 0 71 0 71 72 a 0 71 72 a 0 1 2 1 1 1 0 0 0 0 0 0 Thus 0 is a free variable7 and a cb 720 We can therefore write the intersection in parametric vector form as either cu 7 26V or CW 1 73 0 4 3 Consider the matrico A 0 0 2 71 How many solutions are there to the 0 0 0 75 homogeneous equation Ax 0 Write the solution set as a parametric ueetor equation Row reducing A to reduced echelon form we have 1 73 0 0 1 73 0 0 Aa 0 0 2 0 a 0 0 1 0 0 0 0 1 0 0 0 1 We see that 2 is a free variable so there are in nitely many solutions Writing the basic variables in terms of the free variables we have 1 3x2 x3 0 4 0 which in parametric vector form becomes 1 gig 3 7 2 7 2 7 1 X7 3 7 0 i z 0 4 0 0 4 a Parameterize all solutions of the matrir equation Ai 0 when A Row reducing the associated augmented matrix we get a 732 variable namely zg Setting 3 to be 1 we get the solution 12 All solutions are 1 There is one free 732 in the span of this solution so are parametrized as c 12 Note if there are more 1 free variables set one to be equal to 1 the rest to be 0 to get your basic solutions77 which span all solutions 1 b With A as before parameterize all solutions of the equation A 2 3 Oops didn7t check in proofreading that this should only be a two dimensional vector 1 so lets pretend I said 2 instead Then we only need to nd one solution by letting 3 x3 0 so x2 must be 71 We then nd 1 must be 0 So 1 is a solution and we 0 732 get that all solutions are of the form 1 c 12 0 1 5 For each of the following matrices indicate rst whether its columns are linearly inde pendent or linearly dependent and second whether it is invertible 1 1 2 74 1 0 2 1 71 2 1 1 2 a 0 0 g b 71 72 g c 72 0 0 g d 0 72 1 g e 0 2 1 3 2 3 0 71 0 0 0 0 0 3 a Linearly independent Not invertible not squarel b Linearly independent lnvertible c Linearly dependent includes 0 Not invertible d Linearly dependent free variable Not invertible e Linearly independent no free variables lnvertible 03 De ne the terms Row Echelon Form and Reduced Row Echelon Form If you cannot give a precise de nition give several emamples of each to receive partial credit A matrix is in row echelon form if it satis es the following two conditions a The leading entry ie leftmost non zero entry of each row is to the right of the leading entries of the rows above it b Rows without a leading entry ie rows that are all 07s are at the bottom A matrix is in reduced row echelon form if in addition all leading entries are 1 and all entries above a leading entry are 0 5 Give an eccample of a set of vectors V1 Vn that are linearly independent and a linear transformation T such that the images TV1 TV are linearly depen dent Hint Work backwards starting with a set of linearly dependent vectors and then construct T as a matricc transformation Let T be the zero transformation given by TX 0 for all x Let V be any vector that is not zero The set V is linearly independent but TV is the set 0 which is linearly dependent 8 Find the matrip which represents the linear transformation from R2 to R2 which rst re ects across the line y a then rotates 45 degrees clockwise then re ects across the x apis 1 The rst column of this matrix is given by the image of 0 under this transformation which maps to under the rst re ection then to under rotation and then nally to g22 when re ected across the z axis Similarly the second column of H 9 p the transformation matrix is given by the image of a which you can work out to be 2 2 2 2 ixiQ Q represented by 45 degree clockwise rotation by itselfl So in all you get the matrix This is actually the matrix 1 71 0 1 Consider the matrico A 3 72 0 2 and let T R4 a R3 be the linear transfor 1 72 3 0 mation given by TX AX a Is T onto b Is T one to one Please ewplain your answers and show your work Reducing A to its row echelon form we obtain 1 71 0 1 1 71 0 1 Aa 0 1 0 71 a 0 1 0 71 0 71 3 71 0 0 3 72 a Yes T is onto The row echelon form of A has no rows that are all 0 so we can always solve the equation Ax b for any vector b in the codomain R3 In the textbooks terminology we have a pivot position in in every row b No T is not one to one The variable x4 is free so we have in nitely many solutions to the equation Ax 0 In the textbooks terminology we do not have a pivot position in every column We can also see this by noting that A has more columns that rows which means that the columns of A are linearly dependent 1 1 2 Find the inverse of the matrico A 0 2 1 and use it to solve the matrico equations 0 0 3 A 1 1 Afhforb 0 andh 2 0 3 Using the algorithm from class row reducing A to get I at the same time as performing 1 712 712 the operations on I to get A l we nd that A 1 0 12 716 Solutions 0 0 13 for equations are obtained by multiplying the given vectors by this matrix to get 1 1 1 732 A 1 0 0 in the rst case and A 1 2 12 in the second case 0 0 3 1 11 H D Prove that ifT is a linear transformation then T6 6 One of the two basic properties de ning a linear transformation is that Tei7 eTi7 If we plug in e 0 then the left hand side becomes TOi7 T0 and the righ hand side becomes 0Ti7 0 giving the stated equality Prove that if the vectors 2112 are linearly dependent then so are the vectors 5 2 7amp7 y2 By de nition the fact that 2115 are linearly dependent means that there are 0102033 not all zero such that elf 0217 035 This means then that 01L 7 02f 022 17 032 If the original 0102053 are not all zero then neither are 01L 7 020203 proof if either 02 or 03 is not zero were done Otherwise if they are zero then 01 must not be zero by assumption which means 01 7 02 cl 7 0 is not zero either

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