Linear Algebra with NumPy Primer
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What does this mean?
the set of all real-valued matrices with n rows and d columns
What is this called? What does it mean? 1(x)
• binary indicator function, • returns 1 if x is a true boolean and 0 otherwise
What does this mean?
the set of all real-valued n-dimensional vectors
What does this mean?
the set of real numbers
What is this?
• A pair of input vector and output scalar • The superscript indicates the index of this pair, not the exponent. In other words, x(5) is the input vector at index 5, not x raised to the power of 5.
What is this? |C|
• Determinant if C is a matrix • Cardinality (number of member elements) if C is a set.
What’s a column vector?
Each entry occupies one row.
What’s this?
One use: denotes the i-th entry of a column vector
What does this mean?
Row vector. Each entry covers 1 column
For matrix A, what is
denotes the i-th row of A
For matrix A, what is
denotes the i-th column of A
What can dot product be applied to? What’s the formula?
Two vectors of same dimensions
What’s the inner product?
Same as dot product
• What can outer product be applied to? • What does it result in? • What’s the formula?
• Vectors (can have different number of elements)
What’s another way to express a dot product for two column vectors x and y?
What’s this for two vectors x and y?
The outer product
What’s a matrix-vector product?
• What are the properties of matrix multiplication? (2) • What properties does it not have? (1)
Matrix multiplication is: • associative, (AB)C=A(BC) • distributive, A(B+C)=AB+AC Is not: • commutative, AB≠BA
What’s a property of transposes?
What’s the Hadamard product?
elementwise matrix multiplication
What does this mean?
A standard basis vector - ith entry is 1 and the other entries are all 0
What does this denote?
Identity matrix of nxn size
What’s the point of a basis vector?
By multiplying a basis vector with a vector x∈Rn, we obtain the single entry xi
Describe the idea of one-hot encoding
• In each row of the resulting one-hot-encoded matrix: • In each row of the original vector, find the value • This value will serve as the column index of the “1” entry in the same row in the corresponding matrix. • The remaining values in that row are 0