## Transcribed Text

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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