Tag Archives: 基底

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 .

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 .

本週問題是推導兩個座標系統的變換矩陣。 Let and be bases for a subspace in . Let and . Show that the change of coordinates matrix from to is .

在有限維向量空間的任何一個線性獨立向量集都可擴大成為一組基底。 If is a finite-dimensional vector space and if is any set of linearly independent vectors in , prove that, unless already form a basis, we can find vectors so that is a basis.

證明維數定理：一個有限維向量空間的任兩組基底有相同的元素數。 Prove that the number of elements in any basis of a finite-dimensional vector space is the same as in any other basis.

利用內積空間的基底表達兩個向量的內積。 If is an orthonormal basis for an inner-product space , show that for every .

本週問題是尋找函數子空間 的基底。 Let be an vector space of polynomials of degree at most . Find a basis for the subspace of all polynomials in such that .