Fundamental of Algebra:
Linear algebra is a branch of mathematics that is widely used throughout science and engineering. A good understanding of linear algebra is needed for understanding of many machine learning algorithms, and work for deep learning algorithms.
Basics of Linear algebra:
1. Scalars, Vectors, Matrices and Tensors
2. Multiplying Matrices and Vectors
3. Identity and Inverse Matrices
4. Linear Dependence and Span
5. Norms
6. Special Kinds of Matrices and Vectors
7. Eigen decomposition
8. Singular Value Decomposition (SVD)
9. The Moore Penrose Pseudoinverse
10. The Trace Operator
11. The Determinant
Linear
Algebra:
Linear
algebra is a branch of mathematics that is widely used throughout science and
engineering. A good understanding of linear algebra is needed for understanding
of many machine learning algorithms, and work for deep learning algorithms.
Basics
of Linear algebra:
1. Scalars, Vectors,
Matrices and Tensors
2. Multiplying Matrices and Vectors
3. Identity and Inverse Matrices
4. Linear Dependence and Span
5. Norms
6. Special Kinds of Matrices and Vectors
7. Eigen decomposition
8. Singular Value Decomposition (SVD)
9. The Moore Penrose Pseudoinverse
10. The Trace Operator
11. The Determinant
1. Scalars, Vectors, Matrices and Tensors
Scalars:
A scalar is just a single number, in contrast to most of the other objects
studied in linear algebra, which are usually arrays of multiple numbers. We
write scalars in italics. We usually give scalars lower-case variable names.
When we introduce them, we specify what kind of number they are.
Example-1,
we might say “Let s ∈
R be the slope of the line,” while defining a real-valued scalar, or “Let n ∈ N be the number of
units,” while defining a natural number scalar.
Vectors: A
vector is an array of numbers. The numbers are arranged in order. By using its index,
we can identify each and every individual number from array. Naturally, vectors
represent with lower case names written in bold example x.
The
elements of the vector are identified by writing its name in italic typeface,
with a subscript. The first element of x is x1, the second element
is x2 and so on. We also need to say what kind of numbers are stored
in the vector. If each element is in R, and the vector has n elements, then the
vector lies in the set formed by taking the Cartesian product of R n times,
denoted as R n . When we need to explicitly identify the elements of a vector,
we write them as a column enclosed in square brackets:
Sometimes
we need to index a set of elements of a vector. In this case, we define a set
containing the indices and write the set as a subscript. For example, to access
x1, x3 and x6, we define the set S = {1, 3, 6} and write xS.
Sometimes
we need to index a set of elements of a vector. In this case, we define a set
containing the indices and write the set as a subscript. For example, to access
x1, x3 and x6, we define the set S = {1, 3, 6} and write xS..
Matrices:
A matrix is a 2-D array of numbers, so each element is identified by two
indices instead of just one. We usually give matrices upper-case variable names
with bold typeface, such as A. If a real-valued matrix A has a height of m and
a width of n, then we say that A ∈
Rm×n . For example, A1,1 is the upper left entry of A and Am,n is
the bottom right entry. The transpose of the matrix can be thought of as a
mirror image across the main diagonal.
