Part I (必做题，共5题，70分)
Let denote the t of all real polynomials of degree less than 3 with domain（定义域） . The addition and scalar multiplication are defined in the usual way. Define an inner product on by
.
(1) Construct an orthonormal basis for from the basis by using the Gram-Schmidt orthogonalization process.
(2) Let. Find the projection of onto the subspace spanned by{}.
Solution:
(1) 画蛇添足读后感 吃莲藕有什么好处, ,
, , 退伍费一览表
,
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(2)
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Let be the linear transformation on (the vector space of real polynomials of degree less than 3) defined by
.
(1) Find the matrix reprenting with respect to the ordered basis [] for .
(2) Find a basis for such that with respect to this basis, the matrix B reprenting is diagonal.
(3) Find the kernel（核） and range （值域）of this transformation.
Solution:
(1)
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(2)
(The column vectors of T are the eigenvectors of A)
The corresponding eigenvectors in are
(T diagonalizes A)
. With respect to this new basis , the reprenting matrix of is diagonal. ------------------------------------------------------------------------------------------------------------------- (3) The kernel is the subspace consisting of all constant polynomials.
The range is the subspace spanned by the vectors
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Let .
(1) Find all determinant divisors and elementary divisors of.
(2) Find a Jordan canonical form of .
(3) Compute . (Give the details of your computations.)
Solution:
(1)
,(特征多项式 . Eigenvalues are 1, 2, 2.)
Determinant divisor of order , ,
Elementary divisors are .
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(2) The Jordan canonical form is
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(3) For eigenvalue 1, ， An eigenvector is
For eigenvalue 2, , An eigenvector is
Solve , we obtain that
,
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Suppo that and .
(1) What are the possible minimal polynomials of? Explain.
(2) In each ca of part (1), what are the possible characteristic polynomials of? Explain.
Solution:
(1) An annihilating polynomial of A is .
The minimal polynomial of A divides any annihilating polynomial of A.
The possible minimal polynomials are
, , and .
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(2) The minimal polynomial of A divides the characteristic polynomial of A. Since A is a matrix of order 3, the characteristic polynomial of A is of degree 3. The minimal polynomial of A and the characteristic polynomial of A have the same linear factors.
Ca , the characteristic polynomial is
Ca , the characteristic polynomial is
Ca 椒太郎, the characteristic polynomial is or
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顽皮的杜鹃歌曲
Let . Find the Moore-Penro inver of .
Solution:
,
也可以用SVD求.
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Part II (选做题, 每题10分)
请在以下题目中（第6至第9题）选择三题解答. 如果你做了四题，请在题号上画圈标明需要批改的三题. 否则，阅卷者会随意挑选三题批改，这可能影响你的成绩.
第6题 Let be the vector space consisting of all real polynomials of degree less than 4 with usual addition and scalar multiplication. Let be three distinct real numbers. For each pair of polynomials and in, define
