Vector Space | Definition Of Vector Space | Examples Of Vector Space | Linear Algebra

Summary

Dr. Gajendra Purohit introduces his YouTube channel, dedicated to Engineering Mathematics and BSc, emphasizing its usefulness for competitive exam preparation. In this video, he starts a new topic on vector spaces, a key concept in linear algebra. He highlights the importance of prior knowledge in Group Theory, discussing internal and external composition, vector addition, and scalar multiplication. The video also covers essential properties of vector spaces and includes practical examples to demonstrate whether certain sets can be considered vector spaces. Dr. Purohit provides an overview of upcoming content on subspaces and their properties.

Takeaways

• 😀 Dr. Gajendra Purohit is teaching Engineering Mathematics & BSc topics, focusing on vector spaces and linear algebra.
• 😀 His YouTube channel is helpful for students preparing for competitive exams involving higher mathematics.
• 😀 A new series on CSIR NET is also available for students in life sciences, physics, mathematics, and more, focusing on Part A (general aptitude).
• 😀 Understanding vector spaces is crucial as linear algebra is built on this concept. Group Theory (including concepts like Group, Ring, and Field) is foundational before studying vector spaces.
• 😀 Internal and external composition are key concepts in vector spaces. Internal composition means that operations between two vector elements should return another vector.
• 😀 The mapping for vector spaces is from the field F and the vector space V, where the operation results in a vector element, maintaining closure.
• 😀 Vector addition should satisfy certain properties: closure, commutativity (abelian), associativity, the identity element (0), and invertibility of vectors.
• 😀 Scalar multiplication in a vector space must also follow certain properties, such as closure, distributivity, and maintaining the identity element of the field.
• 😀 Example: Q(Z) is not a vector space because Z is not a field and does not have a multiplicative inverse, which is required in a vector space.
• 😀 A typical exam question involves proving whether a given set of n-tuples is a vector space. It involves checking if the set satisfies the properties of abelian groups and scalar multiplication.
• 😀 The proof for n-tuples involves checking the properties of closure, commutativity, and existence of identity and inverse elements in addition, as well as the distributive properties of scalar multiplication.

Q & A

What is a vector space?

A vector space is a set of vectors that can be added together and multiplied by scalars from a field. It follows specific properties, such as closure under addition, commutativity, and the existence of an additive identity and inverse.

What is the difference between internal and external composition in vector spaces?

Internal composition refers to the operation where the result of applying an operation (like vector addition) between two vectors still lies within the vector space. External composition involves multiplying a vector by a scalar from a field, and the result should also be a vector within the same space.

What are the properties that define a vector space?

The key properties of a vector space include: closure under addition, commutativity, associativity, the existence of an additive identity (zero vector), and the existence of additive inverses. Additionally, scalar multiplication must be distributive and associative.

Why must a field be used in defining vector spaces?

A field is required because it provides the necessary structure for scalar multiplication. For a set to be a vector space, the scalars must come from a field to ensure properties like the existence of multiplicative inverses and distributivity hold.

Can you explain why Q(Z) is not a vector space?

Q(Z) is not a vector space because Z (the integers) is not a field. A field requires multiplicative inverses for all non-zero elements, but integers do not satisfy this condition (e.g., there is no multiplicative inverse for 2 in Z). Thus, Q(Z) cannot be a vector space.

What is meant by closure under addition in vector spaces?

Closure under addition means that if you take any two vectors from the vector space and add them together, the result will still be a vector within the same vector space. This is one of the fundamental properties that defines a vector space.

What is the additive identity in a vector space?

The additive identity is the zero vector, which is a vector where all components are zero. When added to any other vector in the space, it does not change the original vector (i.e., v + 0 = v).

What is the significance of scalar multiplication in vector spaces?

Scalar multiplication is a key operation in vector spaces. It involves multiplying a vector by a scalar from a field. This operation must satisfy properties like distributivity (both over vector addition and scalar addition), and the result must be another vector within the space.

How can we prove if a set of n-tuples is a vector space?

To prove that a set of n-tuples is a vector space, we need to verify that the set satisfies the properties of an abelian group under vector addition, and that scalar multiplication follows the required rules (distributivity, associativity, etc.). If these properties hold, the set of n-tuples is a vector space.

Why is commutativity important in vector spaces?

Commutativity is important in vector spaces because it ensures that the order of vector addition does not affect the result. For any two vectors **v₁** and **v₂**, the sum should be the same regardless of their order: **v₁ + v₂ = v₂ + v₁**.