Category Archives: pow 向量空間

每週問題 June 5, 2017

證明 是不可逆矩陣的一個充分條件。 Let and be matrices, where is an odd number. Prove that if then at least one of the matrices and is singular. Advertisements

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每週問題 May 8, 2017

這是關於基底的一個充分條件問題。 Let be vectors in () such that for . Prove that any of these vectors form a basis of .

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每週問題 April 24, 2017

證明矩陣積的值域與零空間的維數恆等式。 Let be an matrix and be an matrix. Prove that . Note that and denote the column space and nullspace of , respectively.

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每週問題 December 19, 2016

線性相關的向量集的一道線性組合問題。 Let be a vector space, , and let . Prove that if , then there exist scalars not all of them equal to zero such that and .

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每週問題 October 24, 2016

對於任一複矩陣 , 總是成立嗎? Let be an complex matrix. Prove or disprove the following statements. (a) . (b) .

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每週問題 August 15, 2016

為甚麼 是一個線性相關集? Let be the set containing only the zero vector. (a) Explain why must be linearly dependent. (b) Explain why the empty set is a basis for .

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每週問題 August 8, 2016

這是生成空間的一個等價性質。 For a set of vectors , prove that is the intersection of all subspaces that contain .

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每週問題 May 30, 2016

一個線性方程的解集合所包含的最大線性獨立向量數是多少? Let be an matrix and be the solution set for a consistent system of linear equations for some . (a) If is a maximal independent subset of and is any particular solution, show that , where denotes the nullspace … Continue reading

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每週問題 May 23, 2016

這是關於矩陣秩的擾動問題。 Let and be matrices. If and , show that . In words, a perturbation of rank can change the rank by at most .

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每週問題 May 16, 2016

證明兩個矩陣的秩差的不等式。 Let and be matrices. Show that .

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