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  • Linear Algebra Lecture5 note

    Section 2.7     PA=LU

    and Section 3.1   Vector Spaces and Subspaces

     


    Transpose(转置)

    image

    example:

    image

    特殊情况,对称矩阵(symmetric matrices),例如:

    image

    image

    思考:R^R(R的转置乘以R)有什么特殊的?

    回答:always symmetric

    image

    why?

    image

     


    Permutation(置换)

    P=execute row exchanges

    image

    之前A=LU是建立在no row exchanges 的基础上的,但不可能每一个矩阵都是完美的,有些矩阵需要通过行变换处理,

    即PA=LU (any invertible A)

    P= indentity matrix with reordered rows

    置换矩阵是重新排列了的单位矩阵

    counts reorderings(counts all the n * n permutations : n!

    性质:

    image

     


    Vector Spaces

    Example:

    R^2= all 2 dimensional real vectors = “x-y”plane,

     image

    R^3= all vectors with 3 component

    image

    R^n = all vectors with n component

    思考:not  a  vector space? what’s the condition?

    回答:向量空间必须对数乘和加法两种运算是封闭的(线性组合封闭)

    比如说,二维平面子空间  line in R^2 through zero vector

    总结:

    subspaces of R^2: all of R^2(itself), any line through zero vector (L), zero vector only (Z)

    subspaces of R^3: all of R^3(itself), any plane through zero vector (P), any line through zero vector (L), zero vector only (Z)

    example:

    image

    cols in R^3, all their combinations form a subspace called column space, C(A)

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  • 原文地址:https://www.cnblogs.com/nanocare/p/6007374.html
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