Definition of Basis: A subset \(B\) of a vector space \(V\) is said to be a basis if every vector in \(V\) can be expressed uniquely as a linear combination...
Definition of Basis: A subset (B) of a vector space (V) is said to be a basis if every vector in (V) can be expressed uniquely as a linear combination of vectors in (B).
Key Points:
A basis should be linearly independent, meaning no vector in (B) can be expressed as a linear combination of vectors in (B) other than the zero vector.
A basis should be maximal, meaning no proper subset of (B) is a basis.
A basis can be finite or infinite. A finite basis has a finite number of vectors, while an infinite basis has an infinite number of vectors.
The elements of a basis are linearly independent, meaning no vector in the basis can be expressed as a linear combination of other vectors in the basis.
The dimension of a vector space (V) is equal to the number of elements in a basis for that space