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無年度 - 主題課程_線性映射:基底變換和座標變換#107933
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題組內容
1. True or False (25%. 5 pts each) For each of the statements that follows, answer true if the statement is always true and false otherwise.
(d) If N(A) = {0}, then the system Ax = b will have a unique least squares solution.
其他申論題
(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
(e) If{u1, u2,... ,uk} is an orthonormal set of vectors inRnand U = {u1, u2,... ,uk} then UUT = In (the n x n identity matrix).
#462719
(15%) Let T:P2(R)> P2(R) be a linear operator that is defined according to T(g(x))=g(x)+ , in which P2(R) is a set of all polynomials with real-value coefficients and degree n, n=0,1,2. Please find the eigenvalues and the associated eigenvectors of the operator
#462720
a) One solution.(7%)
#462721
b) No solution.(7%)
#462722
c) Infinitely many solutions. (6%)
#462723
Consider the following linear equation systems, express these equations as the matrix form Ax=b. Then find the solution of the vector x. (5%)
#462724