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11. (10%) Let Q be an n✖ n matrix. Then which of the following set is not a subspace? (A) Col O (B) Null Q (C) rank Q (D) Row Q (E) None of the above
.(5%) Determine the rank of the matrix in the following. (A)0 (B)1 (C)2 (D)3 (E) 4
Problem 1. (10 %) Given a matrix where a is a real number. Find the rank of and give the corresponding range of a.
(10%) A= , find bases for Nul A and Col A.
(10%) The nullities of the matrices BBT - λ/ for λ= 0, 1,2,3, 4 are______,________,_________,_______ respectively.
(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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(A)0
(B)1
(C)2
(D)3
(E) 4
and give the corresponding range of a.
, find bases for Nul A and Col A.
{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.