tut4 - Math 136 Tutorial 4 Problems 12 3 1 3 1: Let A = 3...

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Unformatted text preview: Math 136 Tutorial 4 Problems 12 3 1 3 1: Let A = 3 −1, B = ,C= . −1 −1 4 21 Calculate the following or state that they don’t exist: a) AB b) CB c) C T AT . 2: Let S = {(1, 1, −1), (0, 1, −1), (2, −1, 1), (−1, 2, 3)}. a) Write the system of linear equations we would need to solve if we wanted to determine if S is linearly independent or dependent. b) Just by looking at the system of linear equations, how can you tell that S must be linearly dependent? c) By solving the system, find two linear combinations of the vectors in S that proves that S is linearly dependent. 3: Let A be an m × n matrix. Prove that the system of linear equations Ax = b has a solution for every b ∈ Rm if and only if the rank of A = m. 4: Let S = {v1 , v2 , v3 } be a set of vectors in R3 . Prove that span S = R3 if and only if S is linearly independent. 1 ...
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