Category Archives: pow 向量空間

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 .

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.

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 .

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) .

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 .

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 .

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 →

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 .

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