(5 points) Defining the a1, a2,...，an being n column vectors in, R2 and let A = [a1 a2 ... an],(1)the vectors a1，a2，...， an will be linearly independent if and only if A is nonsingular. (2) the detcrminate of A is nonzero. (3) the vectors a1, a2, an will be linearly independent if and only if the null space N(A) ={0}.(4) the rank of A plus the nullity of A equals n.(5) the rank of ATequals the rank of A.

(13%) Given a vector space V over F. Define the dual space of V* of V as the set of all functions (also known as linear functionals) from V to F, i.e, V* {f|f : V →F}. It is obvious that V* is itself also a vector space with the addition + :V*'x V*→ V* and scalar multiplication * : F x V*→ V* defined as pointwise addition as well as pointwise scalar multiplication. Given any linear transformation T : V→ W. The transpose Tt is a linear transformation from W* to V* defined by Tt(f) = fT for any f W*. For every subset S of V, we define the annihilator Suppose V, W are both finite-dimensional vector spaces and T : V → W is linear. Prove that .N(Tt) = (R(T))0.

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