斯坦福大學(xué)機(jī)器學(xué)習(xí)課程講義第三講-線性代數(shù)基礎(chǔ).pptx

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1、Linear Algebra review (optional),Matrices and vectors,Machine Learning,,Dimension of matrix: number of rows x number of columns,Matrix: Rectangular array of numbers:,Matrix Elements (entries of matrix),“ , entry” in the row, column.,Vector: An n x 1 matrix.,n-dimensional vector,1-indexed vs 0-inde

2、xed:,element,Linear Algebra review (optional),Addition and scalar multiplication,Machine Learning,,Matrix Addition,Scalar Multiplication,Combination of Operands,Linear Algebra review (optional),Matrix-vector multiplication,Machine Learning,,Example,Details:,,,,m x n matrix (m rows, n columns),n x 1

3、matrix (n-dimensional vector),m-dimensional vector,To get , multiply s row with elements of vector , and add them up.,Example,House sizes:,Linear Algebra review (optional),Matrix-matrix multiplication,Machine Learning,,Example,Details:,,,,m x n matrix (m rows, n columns),n x o matrix (n rows, o co

4、lumns),m x o matrix,The column of the matrix is obtained by multiplying with the column of . (for = 1,2,,o),Example,House sizes:,Matrix,Matrix,Have 3 competing hypotheses:,1.,2.,3.,Linear Algebra review (optional),Matrix multiplication properties,Machine Learning,,E.g.,Let,Let,Compute,Compute,I

5、dentity Matrix,For any matrix ,,Denoted (or ). Examples of identity matrices:,Linear Algebra review (optional),Inverse and transpose,Machine Learning,,Not all numbers have an inverse.,Matrix inverse: If A is an m x m matrix, and if it has an inverse,,Matrices that dont have an inverse are “singular” or “degenerate”,Not all numbers have an inverse.,Matrix inverse: If A is an m x m matrix, and if it has an inverse,,Matrices that dont have an inverse are “singular” or “degenerate”,Matrix Transpose,Example:,Let be an m x n matrix, and let Then is an n x m matrix, and,