Echelon form and reduced echelon form are row equivalent to the original form.
Span{v1, v2, v3,...... vp} is the collection of all vectors that can be written in the form c1*v1 + c2*v2 + ...... cp*vp with c1, .... cp scalars.
Ax = 0 has a nontrival solution if and only if the equation has at least one free variable.(not full column rank)
Ax = b 的解等于 Ax = 0 和 特解的和。
解线性方程组流程P54。
线性无关指任何向量不能组合成其中一个向量。
Ax = b : ColA1 * x1 + ColA2 * x2 +.... ColAm * xm = b
Matrix Transformations: T(x) = Ax is linear transformation.
转换矩阵是各维单位转换后的组合。A = [T(e1) T(e2) .. T(en)]
A mapping T: R^n -> R^m is said to be onto R^m if each b in R^m is the image of at least one x in R^n. (Ax = b 有解)
A mapping T: R^n -> R^m is said to be one-to-one R^m if each b in R^m is the image of at most one x in R^n.
第二节:Matrix Operation
Each column of AB is a linear combination of the columns of A using weightings from the corresponding columns of B. AB = A[b1 b2 b3 b4 ,,, bp] = [Ab1 Ab2 ... Abp]
Each row of AB is a linear combination of the columns of B using weightings from the corresponding rows of A.
Warning: AB != BA. AB = AC !=> B = C. AB = 0 !=> A = 0 or B = 0
逆矩阵的定义:A-1*A = A*A-1 = E. 可以推导出A为方阵,详见Exercise 23-25 ,Section 2.1. A可逆的充要条件为A满秩(行列式不等于0)。
The dimension of a nonzero subspace H, denoted by dim H, is the numbers of vectors in any basis for H. The dimension of the zero subspace {0} us defined to be zero.
第三节:Introduction to Determinants
determinant的定义和计算方式。
行消元不改变行列式值。交换行改变正负号。某一行乘以k,那么行列式乘以k。
三角矩阵的行列式为对角元素的乘积。
det(AB) = det(A) * det(B)。
Let A be an invertible n*n matrix. For any b in R^n, the unique solutionx of Ax = b has entries given by xi = det Ai(b)/det(A)。 Ai(b) 表示用b替换A的第i行。
由5可以推导出A^-1 = 1/det(A) * adj A. adj A = [(-1)^i+j* det(Aji)]
An indexed set {v1, v2, ... ... vp} of two or more vectors, with vi != 0, is linearly dependent, if and only if some vj (with j > 1) is a linear combination of the preceding vectors.
Elementary row operation on a matrix do not affect the linear dependence relations among the columns of the matrix.
Row operations can change the column space of a matrix.
x = Pb [x]b: we call Pb the change-of-coordinates matrix from B to the standard basis in R^n.
Let B and C be bases of a vector space V. Then there is a unique n*n matrix P_C<-B such that [x]c = P_C<-B [x]b. The columns of P_C<-B are the C-coordinate vectors of the vectors in the basis B, that is P_C<-B = [[b1]c [b2]c ... [bn]c]. [ C B ] ~ [ I P_C<-B]
第五节:Eigenvectors and Eigenvalues
\(Ax =\lambda * x\)
不同特征值对应的特征向量线性无关。
det(A - λ *I) = 0. 因为(A - λ *I)有非零解。
A is similar to B if there is an invertible matrix P such that P^-1AP = B. They have same eigenvalues.
Suppose A = PDP^-1, where D is a diagonal n*n matrix. If B is the basis for R^n formed from the columns of P, then D is the B-matrix for the transformation x ->Ax. 当坐标系转换为P时,转换矩阵对应变成对角矩阵。