Vector Addition and Scalar Multiplication
Vector addition and scaler multiplication are two main concept in vector space that are explained below:
Vector Addition
When you add two vectors, you add their corresponding components. For example, if you have two vectors v = ⟨v1, v2, v3⟩and w = ⟨w1, w2, w3⟩ their sum v+ wv+ w is ⟨v1+w1, v2+w2, v3+w3⟩. Geometrically, vector addition represents the process of moving one vector’s endpoint to the other vector’s endpoint, forming a new vector from the initial point of the first vector to the final point of the second vector.
Scalar Multiplication
Scalar multiplication involves multiplying a vector by a scalar. For example, if you have a vector ⟨v = v1, v2, v3⟩ and a scalar k, then the scalar multiple kvis ⟨kv1, kv2, kv3⟩. Geometrically, scalar multiplication stretches or compresses the vector without changing its direction, depending on whether the scalar is greater than 1 or between 0 and 1
Linear Combinations and Span
Let v1, v2,…, vr be vectors in Rn . A linear combination of these vectors is any expression of the form
k1v1 + k2v2 + ……… + krVr
where the coefficients k1, k2,…, kr are scalars.
Vector Space- Definition, Axioms, Properties and Examples
A vector space is a group of objects called vectors, added collectively and multiplied by numbers, called scalars. Scalars are usually considered to be real numbers. But there are few cases of scalar multiplication by rational numbers, complex numbers, etc. with vector spaces.
In this article, we have covered Vector Space Definition, Axions, Properties and others in detail.
Table of Content
- What is Vector Space?
- Vector Space Axioms
- Vector Space Examples
- Dimension of a Vector Space
- Vector Addition and Scalar Multiplication
- Vector Space Properties
- Subspaces
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