5. State whether each of the following statements is always) true or (sometimes) false. If it is true, give a brief just

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5 State Whether Each Of The Following Statements Is Always True Or Sometimes False If It Is True Give A Brief Just 1 (31.77 KiB) Viewed 36 times

5. State whether each of the following statements is always) true or (sometimes) false. If it is true, give a brief justification; if it is false, give a numerical counterexample with an explanation. a) If A is a 2 x 4 matrix with two pivot positions, then Ax=b has at least one solution for every DER b) If A is a 2 x 4 matrix and b is a vector in R such that Ax = b has a solution, then A : has two pivot positions. c) If A is a 2 x 5 matrix whose columns span R, then A has three free variables. d) A set of 3 vectors in R2 must span R2. (Hint: 1 e) Let u,v,w be vectors in R". If {v}, {u, w} and {v, w} are each linearly independent, then {u, v, w} is linearly independent. 1) Let A be a matrix such that A = 0. Then Aer is a linear combination of the first three columns of A. (Hint: 1 6) Let vi. Va, vi be vectors in R. The set {V1, V, Va} is linearly independentif = = 0 =0 is a solution to Vi + V2 + GV= 0. (Hint: