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Elem Linear Algebra

by: Henderson Lind II

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3

Elem Linear Algebra MATH 341

Henderson Lind II
UO
GPA 3.8

Jonathan Comes

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Jonathan Comes
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This 3 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 Jonathan Comes in Fall. Since its upload, it has received 15 views. For similar materials see /class/187181/math-341-university-of-oregon in Mathematics (M) at University of Oregon.

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Date Created: 09/08/15
Solutions to problems from section 17 0 0 73 2 Determine if the vectors 0 5 4 are linearly independent 2 78 1 a 00 Solution Consider the matrix A whose columns are the vectors listed above The vectors above are linearly independent if and only if the equation Ax 0 has only the trivial solution which happens if and only if the matrix A has a pivot position in each column otherwise there would be free variables Well 0 0 73 78 1 A 0 5 4 R1153 0 4 2 78 1 0 0 0 73 has a pivot position in each column thus the vectors 0 5 4 are linearly 78 1 independent 4 73 0 i i i 0 712 4 i i Determine if the columns of the matrix 1 0 3 are linearly independent 5 4 6 Solution Using the same argument as in exercise 2 we just need to check whether or not the matrix above has a pivot position in each column 4 73 0 1 0 3 NR 7R 4R 1 0 3 1 0 3 0 712 4 Ring 0 712 4 milligsi 0 712 4 Ring 0 73 712 1 0 3 4 73 0 0 73 712 0 712 4 5 4 6 5 4 6 0 4 76 0 4 76 1 0 3 1 0 3 1 NR3R312R2 22 0 3 NR27 R3 O 1 4 NR4R474R2 0 1 4 NR4R4ER3 0 4 A A A 0 712 4 0 0 16 0 0 39 i 0 4 76 J i 0 0 722 i i 0 0 0 i 4 73 0 i i H i i 0 712 4 Since there s a pivot position in each column we see the columns of the matrix 1 0 3 5 4 64 are linearly independent 1 73 3 72 Determine if the columns of the matrix 73 7 71 2 are linearly independent 0 1 74 3 Solution Since there are at most 3 pivot positions in a 3 x 4 matrix we see that there cannot be a pivot position in every column of the matrix above Thus the columns must not be linearly independent 1 72 2 10 Let V1 75 7 V2 10 7 and V3 79 73 6 h a For what values of h is V3 6 SpanV17 V2 7 b For what values of h is V17 V27 V3 linearly dependent Solution We can answer both questions by row reducing the matrix whose columns are V17 V27 and V3 as follows 1 72 2 gig sgl 1 72 2 75 10 79 31131 0 0 1 73 6 h 0 0 h 6 1 72 l 2 The second row ofthe reduced matrix above tells us that the augmented matrix 75 10 l 79 73 6 l h corresponds to an inconsistent system regardless to h Thus the vector equation 901V1 2V2 V3 never has a solution7 which is the same as saying that V3 is never in SpanV17V2 This answers part a To answer part b notice that the second column of the reduced matrix above is not a pivot column regardless of h Thus the vectors V17 V27 and V3 are always linearly dependent 1 72 0 20 Determine by inspection whether the vectors 4 7 5 7 0 are linearly indepen 77 3 0 dent Justify your answer Solution Notice that 1 72 0 0 0 4 0 5 1 0 0 77 3 0 0 so that 1 07 2 O7 and 3 1 is a nontrivial solution to the equation 1 72 0 0 m1 4 m2 5 3 0 0 77 3 0 0 72 0 Thus the vectors 4 7 5 7 0 are not linearly independent 77 3 0 22 True or False Justify your answer a Two vectors are linearly dependent if and only if they lie on a line through the origin Solution TRUE We noticed in class and on page 67 that two vectors are linearly dependent if and only if one of the vectors is a scalar multiple of the other But a vector is a scalar multiple of another vector if and only if the two vectors lie on the same line through the origin 32 Given A 3 00 b If a set contains fewer vectors than there are entries in the vectors then the set is linearly dependent Solution FALSE Here s a counter example The set 1 contains one vector with 2 entries but is linearly independent because the only one element sets which are linearly dependent are the one element sets containing the origin 0 If x and y are linearly independent and if z is in the Spanx y then x y z is linearly dependent Solution TRUE 2 E Spanxy implies that we can nd real numbers a and b with ax by z Subtracting the vector 2 from each side we get ax by 7 z 0 so that 01 a 02 b 03 71 is a nontrivial solution to the equation 01x 1 02y 032 0 Thus x y z is linearly dependent CL If a set in R is linearly dependent then the set contains more vectors then there are entries in each vector Solution FALSE Here s a counter example actually I m giving an in nite number of counter examples one for each n 1 2 3 The one element set 0 C R is always linearly dependent 4 1 77 5 73 the third column Find a nontrivial solution of Ax 0 6 3 observe that the rst column plus twice the second column is 3 Solution Rewriting the given vector equation 4 1 6 4 1 6 0 77253ltgt7725730 9 73 3 9 73 3 0 as a matrix equation we get 1 A 2 0 71 Thus a nontrivial solution of Ax 0 is x 2 71 True or false 7 If V1 V2 V3 V4 are linearly independent vectors in R4 then V1 V2 V3 is also linearly independent Solution TRUE Any nontrivial solution a 60 to the equation z1V1 ngz m3V3 0 would give us the nontrivial solution a b c 0 to the equation 901V1 2V2 3V3 4V4 0 Since V1 V2 V3 V4 are linearly independent no such nontrivial solution can exist by the de nition of linear independence Thus again by de nition of linear independence V1 V2 V3 is also linearly independent

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