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![Page 1: Linear Algebra review (optional) - · PDF fileLinear Algebra review (optional) Inverse and transpose Machine Learning Andrew Ng Not all numbers have an inverse. Matrix inverse: If](https://reader033.vdocuments.net/reader033/viewer/2022051722/5aa7eef37f8b9a54748ca8ca/html5/thumbnails/1.jpg)
Andrew Ng
Linear Algebra review (optional)
Matrices and vectors
Machine Learning
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Andrew Ng
Dimension of matrix: number of rows x number of columns
Matrix: Rectangular array of numbers:
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Andrew Ng
Matrix Elements (entries of matrix)
“ , entry” in the row, column.
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Andrew Ng
Vector: An n x 1 matrix.
1-indexed vs 0-indexed:element
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Andrew Ng
Linear Algebra review (optional)
Addition and scalar multiplication
Machine Learning
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Andrew Ng
Matrix Addition
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Andrew Ng
Scalar Multiplication
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Andrew Ng
Combination of Operands
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Andrew Ng
Linear Algebra review (optional)
Matrix-vector multiplication
Machine Learning
![Page 10: Linear Algebra review (optional) - · PDF fileLinear Algebra review (optional) Inverse and transpose Machine Learning Andrew Ng Not all numbers have an inverse. Matrix inverse: If](https://reader033.vdocuments.net/reader033/viewer/2022051722/5aa7eef37f8b9a54748ca8ca/html5/thumbnails/10.jpg)
Andrew Ng
Example
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Andrew Ng
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.
![Page 12: Linear Algebra review (optional) - · PDF fileLinear Algebra review (optional) Inverse and transpose Machine Learning Andrew Ng Not all numbers have an inverse. Matrix inverse: If](https://reader033.vdocuments.net/reader033/viewer/2022051722/5aa7eef37f8b9a54748ca8ca/html5/thumbnails/12.jpg)
Andrew Ng
Example
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Andrew Ng
House sizes:
![Page 14: Linear Algebra review (optional) - · PDF fileLinear Algebra review (optional) Inverse and transpose Machine Learning Andrew Ng Not all numbers have an inverse. Matrix inverse: If](https://reader033.vdocuments.net/reader033/viewer/2022051722/5aa7eef37f8b9a54748ca8ca/html5/thumbnails/14.jpg)
Andrew Ng
Linear Algebra review (optional)
Matrix-matrix multiplication
Machine Learning
![Page 15: Linear Algebra review (optional) - · PDF fileLinear Algebra review (optional) Inverse and transpose Machine Learning Andrew Ng Not all numbers have an inverse. Matrix inverse: If](https://reader033.vdocuments.net/reader033/viewer/2022051722/5aa7eef37f8b9a54748ca8ca/html5/thumbnails/15.jpg)
Andrew Ng
Example
![Page 16: Linear Algebra review (optional) - · PDF fileLinear Algebra review (optional) Inverse and transpose Machine Learning Andrew Ng Not all numbers have an inverse. Matrix inverse: If](https://reader033.vdocuments.net/reader033/viewer/2022051722/5aa7eef37f8b9a54748ca8ca/html5/thumbnails/16.jpg)
Andrew Ng
Details:
m x n matrix(m rows,
n columns)
n x o matrix(n rows,
o columns)
m x omatrix
The column of the matrix is obtained by multiplyingwith the column of . (for = 1,2,…,o)
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Andrew Ng
Example
2 7
7
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Andrew Ng
House sizes:
Matrix Matrix
Have 3 competing hypotheses:
1.
2.
3.
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Andrew Ng
Linear Algebra review (optional)
Matrix multiplication properties
Machine Learning
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Andrew Ng
Let and be matrices. Then in general,
(not commutative.)
E.g.
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Andrew Ng
Let
Let
Compute
Compute
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Andrew Ng
Identity Matrix
For any matrix ,
Denoted (or ).Examples of identity matrices:
2 x 23 x 3
4 x 4
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Andrew Ng
Linear Algebra review (optional)
Inverse and transpose
Machine Learning
![Page 24: Linear Algebra review (optional) - · PDF fileLinear Algebra review (optional) Inverse and transpose Machine Learning Andrew Ng Not all numbers have an inverse. Matrix inverse: If](https://reader033.vdocuments.net/reader033/viewer/2022051722/5aa7eef37f8b9a54748ca8ca/html5/thumbnails/24.jpg)
Andrew Ng
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”
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Andrew Ng
Matrix Transpose
Example:
Let be an m x n matrix, and let Then is an n x m matrix, and