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In each case below show that the statement is True or give an example showing that...

In each case below show that the statement is True or give an example showing that it is False.

(i) If {X, Y } is independent in R n, then {X, Y, X + Y } is independent.

(ii) If {X, Y, Z} is independent in R n, then {Y, Z} is independent.

(iii) If {Y, Z} is dependent in R n, then {X, Y, Z} is dependent.

(iv) If A is a 5 × 8 matrix with rank A = 4, then dim(null A) = 3.

(v) If A and B are similar matrices, then so are AT and BT .

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