(b)Find

Let .Show that is linear combination of A and I2. 

Suppose that T: R2→R3 is a linear transformation such that and Determine for any in R2. 

Let T be a linear operator on R3 such that T, Please find the standard matrix of T.

(a) Find .

Let Y be the vector space of 2x2 mattices with real entries, and P3 the vector space of real polynomials of dogree 3 or less. Detine the linear transformation T: V → P3 by == 2a+(6-d)t-(a+c)x2+(a+6-c-d)x3. Find the rank and nullity of T.

(12%) Let V be a finite-dimensional vector space and T :V →V be linear. Suppose rank(T) = rank(T2). Prove that R(T) ∩N(T) = {0}.

(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.

(10%) The nullities of the matrices BBT - λ/ for  λ= 0, 1,2,3, 4  are______,________,_________,_______ respectively. 

(d) (2%) Compute dim(N(Bt A)).

(c) (3%) Compute rank(AtAAAt).

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