Linear algebra vector is one of the most important topics in linear algebra or Algebraic theory. In this linear algebra having some of the sub topics, like vector, matrix and so on. This linear algebra having some more important application in matrices and vector algebra. Now we are going to see about vector algebra and vector spaces in linear algebra theory.
Vector space in Algebra:
A vector space is a concept of a group of vectors or set of vectors. Which are working on two operations, like1.Vector addition and 2. Scalar multiplication. This is general concept of vector. And it is having more subtopics: Vector spaces, sub spaces, fundamental sub spaces, inner product spaces, span, basis and dimension, change of basis, linear independence, least squares, orthogonal matrices, QR- Decomposition, and orthonormal basis.
Example for vector space: the complex number 4 + 5i can be considered a vector space, it is way of explained by
`[[4],[5]]`
the vector space is a “space”, such as abstract objects. Which is we called as vectors. Now we are study about the vectors, and see the vectors like S2, S3… Sn and so on. These all are like a vector spaces.
Vector addition is denoted by = x, y `in` V, (x + y).
Properties of vector algebra:
In this vector addition and scalar multiplication is having some important properties, there are given below,
Vector addition property:
Commutative addition property:` vecx` + `vecy` = `vecy + vecx.`
Associative addition property: `vecx + (vecy + vecz)` = `(vecx + vecy) + vecz` .
Addition identity property: Here it is a 0 vector, so 0 +` vecx ` = `vecx ` for all x.
Addition inverse property: for each x vector, there exist another y vector like`vecx + vecy` = 0.
Scalar multiplication property:
Scalar associative property: α (βx) = (vecαβ) x.
Scalar distributive property: (α + β) x = αx+βx.
Vector distributive property: α(x+y) = αx+αy.
Scalar identity property: 1x = x.
These all are the very important in linear algebra properties in vector.
Types of vector:
Types of Vectors
Zero Vector A vector whose initial and terminal points coincide, is called a zero Vector (or null vector), and denoted as `vec0` . Zero vector can not be assigned a definite direction as it has zero magnitude. Or, alternatively otherwise, it may be regarded as having any direction. The vectors `vec(A A) `, `vec(BB)` represent the zero vectors,
Unit Vector A vector whose magnitude is unity (i.e., 1 unit) is called a unit vector. The Unit vector in the direction of `veca ` given vector a is denoted by `hata` .
Co initial Vectors Two or more vectors having the same initial point are called co initial vectors.
Collinear Vectors Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.
Equal Vectors Two vectors `veca` and `vecb` are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written as `veca = vecb`
Negative of a Vector A vector whose magnitude is the same as that of a given vector (Say, `vec(AB)` ), but direction is opposite to that of it, is called negative of the given vector.
For example, vector `vec(BA)` is negative of the vector `vec(AB)` , and written as `vec(BA) = vec(-AB)`
