Outline (Level 2-3) Linear Space ●Linear Independence o Matrix Rank o Matrix Inverse o Orthogonal Matrix o Range and Null Space o Determinant o Eigenvalue and Eigenvector 25/130
Outline (Level 2-3) Linear Space Linear Independence Matrix Rank Matrix Inverse Orthogonal Matrix Range and Null Space Determinant Eigenvalue and Eigenvector 25 / 130
1.4.Linear Space 1.4.1.Linear Independence For the vector set {x1,x2,..,x,if no vector can be represented as a linear combination of other vectors,then these vectors in the vector set are said to be independent of each other. If n-1 x-∑ax i=1 then vector is dependant on vector set x2...,, Otherwise,vect x is independent of vector set {x1,x2,...,x1. 26/130
1.4. Linear Space 1.4.1. Linear Independence ▶ For the vector set {x1, x2, · · · , xn} , if no vector can be represented as a linear combination of other vectors, then these vectors in the vector set are said to be independent of each other. ▶ If xn = Xn−1 i=1 aixi then vector xn is dependant on vector set {x1, x2, . . . , xn−1} , Otherwise, vect xn is independent of vector set {x1, x2, . . . , xn−1} . 26 / 130
Outline (Level 2-3) Linear Space o Linear Independence o Matrix Rank o Matrix Inverse o Orthogonal Matrix o Range and Null Space o Determinant o Eigenvalue and Eigenvector 27/130
Outline (Level 2-3) Linear Space Linear Independence Matrix Rank Matrix Inverse Orthogonal Matrix Range and Null Space Determinant Eigenvalue and Eigenvector 27 / 130
1.4.2.Matrix Rank The rank of matrix A is the largest number of vectors of independent columns (or rows).It has the following properties: L.ForA∈Rmx,rank(A)≤min(m,n). If rank(4)=min(m,n),then A is full rank; 2.For A E Rmx",rank(A)=rank(AT); 3.ForA∈Rmx",B∈Rmxp,rank(AB)≤min(rank(A),rank(B): 4.ForA,B∈Rmxn,rank(A+B)≤rank(A)+rank(B) 28/130
1.4.2. Matrix Rank ▶ The rank of matrix A is the largest number of vectors of independent columns (or rows). It has the following properties: 1. For A∈R m×n , rank(A)≤min(m, n). If rank(A)=min(m, n), then A is full rank; 2. For A ∈ R m×n , rank(A) = rank(A T ); 3. For A∈R m×n , B∈R n×p , rank(AB)≤min(rank(A), rank(B)); 4. For A, B ∈ R m×n , rank(A + B) ≤ rank(A) + rank(B). 28 / 130
Outline (Level 2-3) Linear Space o Linear Independence o Matrix Rank o Matrix Inverse o Orthogonal Matrix o Range and Null Space o Determinant o Eigenvalue and Eigenvector 29/130
Outline (Level 2-3) Linear Space Linear Independence Matrix Rank Matrix Inverse Orthogonal Matrix Range and Null Space Determinant Eigenvalue and Eigenvector 29 / 130