math54 - quiz 2 - solns

# Math54 quiz 2- - independent What about the rows 1 2 1 1-1 1 The columns of a matrix are linearly independent if and only if the homogeneous system

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Math 54 Quiz 2 Solutions 1. Mark each of the statements below true, false, or nonsensical. No justiﬁcation is needed. (1 point each) (a) A system of equations A x = b is automatically consistent if A has a pivot in every column. FALSE (b) Any 2 × 3 matrix is automatically linearly dependent. NONSEN- SICAL (c) It is not possible for a system of 3 equations and 4 unknowns to have a unique solution. TRUE (d) It is possible to ﬁnd a set of 3 linearly independent vectors in R 4 . TRUE 2. (3 points) Determine if the vector b = 2 - 1 6 is in the span of the three vectors given. a 1 = 1 - 2 0 , a 2 = 0 1 2 , a 3 = 5 - 6 8 . This problem is equivalent to determining if the system with aug- mented matrix 1 0 5 2 - 2 1 - 6 - 1 0 2 8 6 is consistent (has a solution) or not. On doing row reduction on A , we can obtain the matrix 1 0 5 2 0 1 4 3 0 0 0 0 which is consistent (with inﬁnitely many solutions). Thus b is in span { a 1 , a 2 , a 3 } . 1

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3. (3 pts) Determine if the columns of the following matrix are linearly
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Unformatted text preview: independent. What about the rows? 1 2 1 1-1 1 The columns of a matrix are linearly independent if and only if the homogeneous system with the matrix as the coeﬃcients has only the trivial solution. So, we need to solve the system corresponding to the augmented matrix 1 2 0 1 1 0-1 1 0 This matrix is row equivalent to the matrix 1 2 0-1 0 0 0 which represents a system with the unique solution x 1 = 0, x 2 = 0. In particular, the system only has the trivial (all zeros) solution, so the columns are linearly independent. As for the rows of the matrix, they are vectors in ( R ) 2 , and there are three of them, so by a fact stated in class, it is not possible for them all to be linearly independent, so the rows are linearly dependent. 2...
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## This note was uploaded on 09/17/2011 for the course MATH 54 taught by Professor Chorin during the Spring '08 term at University of California, Berkeley.

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Math54 quiz 2- - independent What about the rows 1 2 1 1-1 1 The columns of a matrix are linearly independent if and only if the homogeneous system

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