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State whether the following statements are true or false.
(1) Let V be a vector space. Any set of vectors in V that contains the zero vector
is linearly dependent.
(2) It takes at least three vectors to span R3 .
(3) Every vector space has a unique set of vectors that spans it.
(4) Every subset of a linearly dependent set is linearly dependent.
that D = PT AP is a diagonal matrix.
(1) Find the matrix D.
(2) Find the matrix P.
(3) Find the matrix A5 .
(1) Determine the norm of the vector (3,−1) in this space.
(2) Show that the vectors (2,1) and (−8, 4) are orthogonal in this space.
(3) Determine the distance between the points (3,−1) and (2,5) in this space.
(1) is A diagonalizable? Is A a singular matrix?
(2) is B diagonalizable? Is B a singular matrix?
(Hint: Please explain your answers.)
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