Introduction to Linear Algebra

Many difficult problems can be handled easily when we organize relevant information in a certain way for which we can use Linear algebra.

Linear algebra is the study of vectors and linear functions. Lets deep dive into its basics.

1. Vectors and spaces

• Vectors are things you can add and scalar multiply.

Examples of kinds of vectors:

1.   numbers

2.   n-vectors

3.   2nd order polynomials

4.   polynomials

5.   power series

6.   functions with a certain domain

• A vector space is a set V on which two operations + and · are defined, called vector addition and scalar multiplication.
• A space in any finite number of dimensions, in which points are designated by coordinates (one for each dimension) and the distance between two points.
• Euclidean space is the fundamental space of classical geometry.
• The basic example is n-dimensional Euclidean space  R, where every element is represented by a list of n real numbers, scalars are real numbers, addition is component wise, and scalar multiplication is multiplication on each term separately.
• For a general vector space, the scalars are members of a field F, in which case V is called a vector space over  F .
• Euclidean  n-space  R  is called a real vector space, and  C  is called a complex vector space.
• The operation + (vector addition) must satisfy the following conditions:

Closure: If u and v are any vectors in V, then the sum  u + v  belongs to V.

Commutative law: For all vectors u and v in V,   u + v = v + u

Associative law: For all vectors u, v, w in V,   u + (v + w) = (u + v) + w

Additive identity: The set V contains an additive identity element, denoted by 0, such that for any vector v in V,   0 + v = v  and  v + 0 = v.

Additive inverses: For each vector v in V, the equations   v + x = 0  and  x + v = 0   have a solution x in V, called an additive inverse of v, and denoted by - v.

• The operation (scalar multiplication) is defined between real numbers (or scalars) and vectors, and must satisfy the following conditions:

Closure: If v in any vector in V, and c is any real number, then the product c · v  belongs to V.

Distributive law: For all real numbers c and all vectors u, v in V,  c · (u + v) = c · u + c · v

Distributive law: For all real numbers c, d and all vectors v in V,  (c+d) · v = c · v + d · v

Associative law: For all real numbers c,d and all vectors v in V,  c · (d · v) = (cd) · v

Unitary law: For all vectors v in V,  1 · v = v

2. Matrix Transformations:

Let A be an m×n matrix. The matrix transformation associated to A is the transformation

It multiplies the input matrix A by a scalar x

Let T(x) = Ax, then

In general, if matrix A has n columns v1, v2, …, vn

The values of vector x for some transformations are as given below:

3. Coordinate systems (bases) :Suppose β = {b1, . . . , bn} is a basis for a vector space V and x is in V. The coordinates of x relative to the basis β (or the β− coordinates of x) are the weights c1, . . . , cn such that x = c1b1 + · · · + cnbn. Formula : x = c1b1 + · · · + cnbn.

Co-ordinate vectors:

In this case, the vector in

is called the coordinate vector of x (relative to β), or the β − coordinate vector of x.

Let’s End the mathematical Jargons here and stay tuned for my next post.