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Vector space
Vector space is a mathematical space that encompasses all possible linear combinations of a finite set of vectors. Linear combination: Two vectors are l...
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Vector space is a mathematical space that encompasses all possible linear combinations of a finite set of vectors. Linear combination: Two vectors are l...
Vector space is a mathematical space that encompasses all possible linear combinations of a finite set of vectors.
Linear combination: Two vectors are linearly combined if their corresponding coordinates are simply added together. For example, if we have two vectors, v1 = (1, 2, 3) and v2 = (4, 5, 6), their linear combination v = v1 + v2 = (5, 7, 9).
Vector space: A vector space is a set of vectors that can be expressed as the sum of a finite number of vectors from the base space. For example, the vector space of all 2D vectors can be expressed as the sum of two linearly independent vectors, such as v = (x, y) and w = (x + 1, y - 2).
Basis: A set of linearly independent vectors in a vector space is called a basis. A basis can be chosen such that the vectors in the basis form a spanning set for the space.
Subspace: A subspace is a subset of a vector space that is itself a vector space. In other words, a subspace is a space that can be expressed as the intersection of a finite number of vectors from the original space.
Dimensionality: The dimension of a vector space is the number of linearly independent vectors in a basis. For example, the dimension of the 2D vector space is 2, since any 2D vector can be expressed as a linear combination of two linearly independent vectors.
Examples:
The vector space of all 3D vectors is 3D dimensional.
The vector space of all 1D vectors is 1D dimensional.
The vector space of all polynomials of degree 2 with real coefficients is a 3D dimensional vector space