試題詳解

9. The linear combination of a set of vectors is an essential element in linear algebra. We say a set V is invariant under linear combination if the implication " and any set of vectors the set of all linear combinations V" holds. And we say a mapping L defined on a set X is invariant under linear combination if the form of linear combination is unchanged under L, or more precisely the statement ", and any set of vectors X, the identity = holds" is true. Which one of the following statements related to linear combination is false?
(A) Let S be a subset of a vector space V. Then S is a subspace of V if S is invariant under linear combination.

(B) Let  be a set of k subspaces of a vector space W and denote as the set of all linear combinations of the form , with each chosen freely from . Then spa is also a subspace of W with =

(C) Let A and B be two matrices and denote C := AB. Then each column of C is a linear combination of all columns of A, and so rank≤ rank is implied. 

(D) A mapping L between two vector spaces is a linear transformation if and only if it is invariant under linear combination.
(E) Let (V, (. ,. )V) be an inner product space. Then (. , .)V is invariant under linear combination at either one of its two arguments.