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Lecture 001 on Vector Space

This video is first on the series of teaching you Vector Space for Civil Services (Mains), optional Mathematics. This video introduces you to n-dimensional space in a very subtle way, banking upon your studies you did in you XI an XII standard, and does not allow you to stand confused as to the very abstract topic, i.e. Vector Space. This lesson speaks of the term ‘basis’ and explains you its ‘rough’ meaning without going into the intricacies of the topic.

Lecture 003 on Vector Space

This video introduces you to how to prove a set to be a vector space. This video solves following 01 question.

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Example 01: Let V be the set of all functions from a nonempty set X into a field K. For any functions f, g € V and any scalar k € K, let f + g and kf be the functions in V defined as follows: (f + g)(x) = f(x) + g(x) and (kf)(x) = kf(x), for all x € X. Prove that V is a vector space over K.

Lecture 002 on Vector Space

This video is second on the series of teaching you Vector Space for Civil Services (Mains), optional Mathematics. After a very simple and easy-to-understand introduction as to what are Euclidian spaces, how are Vectors formed, what is ‘basis’, defining Groups etc. this video introduces you to formal definition of Vector Space in a very lucid way, and this video solves following 05 questions also.

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Example 1: Show that the set V of all m × n matrices is a vector space over the field of real numbers.

Example 2: Consider the set V of all n x n real magic squares. Show that V is a vector space over R. Give example of 2 x 2 magic squares. IAS 2020

Example 03: Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated, assume that vector addition and scalar multiplication are the ordinary operations defined on the set.

Example 04: Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated, assume that vector addition and scalar multiplication are the ordinary operations defined on the set.

Example 05: Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated, assume that vector addition and scalar multiplication are the ordinary operations defined on the set.

Lecture 004 on Vector Space

This video introduces you to how to prove a subset of a vector space to be a subspace. In this lecture, the concept of subspace has been explained in a very lucid way giving some examples too.

This video solves following 03 questions.

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Example 1. Examine for subspace for all vectors in R2 whose components are positive or zero.

Example 2. Find out two subspaces from the vector space of 3 by 3 matrices.

Example 3: Let V = R^3. Show that W is a subspace of V where:

(i) W = {(a, b, 0): a, b € R}, i.e. W in the xy plane consisting of those vectors whose third component is 0;

(ii) W = {(a, b, c): a + b + c = 0}, i.e. W consists of those vectors each with the property that the sum of its components is zero.