Category Archives: pow 線性變換

這是一道線性變換的證明問題。 Let and be two vector spaces over the same field. Suppose and are two linear transformations such that for every , is s scalar multiple (depending on ) of . Prove that is a scalar multiple of .

證明一個直覺命題：若一個子空間與線性變換的零空間不交集，則該子空間的像 (image) 的維數等於子空間的維數。 Let and be finite dimensional vector spaces, and be a linear transformation. For a subspace of , the image of under is a subspace of . Prove that if , then . Note that denotes the nullspace … Continue reading →

計算一個線性變換的秩。 Let be an orthonormal set in and be the cross product of and , i.e., . A linear transformation is defined by . Determine the rank of .

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 .

這是判定可逆矩陣的問題。 Let be an matrix. If for any nonzero matrix , show that is nonsingular.

這是關於微分算子的矩陣表示以及特徵值問題。 Let be the differential operator on over defined as follows: If , then . (a) Find the matrix representation of under the basis . (b) Find the eigenvalues of .

這是關於對偶基底的問題。 The vectors , , and form a basis of . If is the dual basis, and if , find , , and .

這是線性泛函的問題。 Define a non-zero linear functional on such that if and , then .

本週問題是證明二同構的向量空間的基底存在唯一的線性映射。 Suppose the vector spaces and have bases and , respectively. Show that there is exactly one linear transformation with the property , for .