(5 points) A nxn matrix A is said to be nonsingular, if (1) A is non-invertible. (2) there exists a multiplicative inverse matix B such that A is row cquivalent to B. (3) the determinate of A is nonzero. (4) Ax=O where x is nonzero vector. (5) A is row equivalent to L

(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.

(5 points) A nxn matrix A is said to be nonsingular, if
(1) A is non-invertible.
(2) there exists a multiplicative inverse matix B such that A is row cquivalent to B.
(3) the determinate of A is nonzero.
(4) Ax=O where x is nonzero vector.
(5) A is row equivalent to L

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