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Matrices and vectors

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

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Matrices and vectors

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  1. Linear Algebra review (optional) Matrices and vectors Machine Learning

  2. Matrix: Rectangular array of numbers: Dimension of matrix: number of rows x number of columns

  3. Matrix Elements (entries of matrix) “ , entry” in the row, column.

  4. Vector: An n x 1 matrix. 1-indexed vs 0-indexed: element n-dimensional vector

  5. Linear Algebra review (optional) Addition and scalar multiplication Machine Learning

  6. Matrix Addition

  7. Scalar Multiplication

  8. Combination of Operands

  9. Linear Algebra review (optional) Matrix-vector multiplication Machine Learning

  10. Example

  11. Details: m x n matrix (m rows, n columns) n x 1 matrix (n-dimensional vector) m-dimensional vector To get , multiply ’s row with elements of vector , and add them up.

  12. Example

  13. House sizes:

  14. Linear Algebra review (optional) Matrix-matrix multiplication Machine Learning

  15. Example

  16. Details: m x n matrix (m rows, n columns) n x o matrix (n rows, o columns) m x o matrix The column of the matrix is obtained by multiplying with the column of . (for = 1,2,…,o)

  17. Example 7 2 7

  18. Have 3 competing hypotheses: House sizes: 1. 2. 3. Matrix Matrix

  19. Linear Algebra review (optional) Matrix multiplication properties Machine Learning

  20. Let and be matrices. Then in general, (not commutative.) E.g.

  21. Compute Let Compute Let

  22. Identity Matrix Denoted (or ). Examples of identity matrices: 2 x 2 3 x 3 4 x 4 For any matrix ,

  23. Linear Algebra review (optional) Inverse and transpose Machine Learning

  24. Not all numbers have an inverse. Matrix inverse: If A is an m x m matrix, and if it has an inverse, Matrices that don’t have an inverse are “singular” or “degenerate”

  25. Matrix Transpose Example: Let be an m x n matrix, and let Then is an n x m matrix, and

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