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Unformatted text preview: Math 136 Sample Term Test 1  2 NOTE :  Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) State the definition of the rank of a matrix. Solution: The rank of a matrix is the number of leading 1s in the reduced row echelon form of the matrix. b) What can you say about the consistency and the number of parameters (free variables) in the general solution of a system of 5 linear equations in 4 variables. Solution: You cannot say anything about the consistency or the number of parameters because the system has more equation than variables and we don’t know the rank of the coefficient matrix. c) Let A = 3 2 1 2 1 4 and B =  2 1 1 1 1 . Calculate AB . Solution: AB = 4 4 5 5 . d) Let S = { ~v 1 ,~v 2 ,~v 3 } be a set of vectors in R 3 . State the definition of the set S being linearly independent. Solution: S is linearly independent if the only solution to c 1 ~v 1 + c 2 ~v 2 + c 3 ~v 3 = ~ 0 is the trivial solution ( c 1 = c 2 = c 3 = 0). e) Give an example of vectors ~u and ~v in R 3 such that the vector equation ~x = s~u + t~v , s,t ∈ R , is not a plane. Solution: We can take ~v = = ~u , then ~x = s~u + t~v is just the origin. 1 2. Consider the system of linear equations: 2 x + 3 y + 3 z = 9 3 x 4 y + z = 5 5 x + 7 y + 2 z = 14 a) Write the augmented matrix and row reduce it to RREF using elementary row operations....
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This note was uploaded on 11/25/2010 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.
 Spring '08
 All
 Math, Linear Algebra, Algebra

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