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Determine whether each of the following statements is true or false. Be sure to justify your answer - if it is true (always true), explain why; if it is false (not always true), explain why or give a counterexample. (a) (3 points) Suppose that {v1, V2, V3, V4} is a basis for a linear subspace V of Rn Then, there exists an orthogonal basis {W1, W2, W3 for V. (b) (3 points) Let V and W be linear subspaces of R51 and consider the set U = {v + W : V E V, W E W}. Then, U is a linear subspace of R51 (c) (3 points) Let v, W E R³ and suppose that V and W are not scalar multiples of each other. If P = span (v, w) and x E R³, then Projp (x) = Projv(x) + Projw (x). Let P be the plane defined by x + 2y + 3z=4. (a) (3 ) Find a point that lies on P (there are many possible answers). (b) For the point you found in part (a), call it V, give a parametric form of a line L in P that passes through V (there are many possible answers). (c) Use the equation for P to check that all points on the line you found in part (b) do indeed lie in P.

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