題組內容

(8%) Let M₂(R) be the vector space of all 2 X 2 matrices with real coefficients and let Pz be the vector space of all real polynomials of degree at most 2. Define T: M₂(R) → P₂ by

=(a+b)+2dx+bx².

Let B=

and C={1，x，x2}，D={1+x，x，1+x²}
Be ordered bases for M₂(R) and P₂, respectively. Determine

(b) (4%) The matrix representation of T relative to ordered bases B, D.

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(a) (4%) The matrix representation of T' relative to ordered bases B, C.

#462711

(a) (3%) Find a basis for V.

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(b) (12%) Find the eigenvalues of T.

#462714

(a) If A and B are n x n matrices that have the same rank, then the rank of A? must equal the rank of B2.

#462715

(b) Let L : R2→ R2 be a linear operator, and let A be the standard matrix representation of L. If L2 is defned by L2(x) = L(L(x)) for all x R2 then L2 is a linear operator and its standard matrix representation is A2.

#462716

(c) If L1 and L2 are both linear operators on a vector space V, then Li + L2 is also a linear operator on V, where L1 + L2 is the mapping defined by (L1+L2)(v)=L1(v)+L2(v)forallvV

#462717

(d) If N(A) = {0}, then the system Ax = b will have a unique least squares solution.

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