Vector Spaces and Subspaces

Basis in a Vector Space/Subspace

• A basis of a vector space/subspace is a set of linearly independent vectors that span the entire space/subspace.
• In other words, a basis is a set of vectors that can be combined to represent every vector in the space/subspace.
• The basis vectors are called linearly independent because none of the vectors can be expressed as a linear combination of the others.

Importance of a Basis

• A basis is important in a vector space/subspace because it allows us to represent every vector in the space/subspace in a unique way.
• This unique representation is essential for performing operations on vectors, such as addition and scalar multiplication.
• A basis also provides a convenient way to specify a vector, as it can be expressed as a linear combination of the basis vectors.

Dimension of a Vector Space

• The dimension of a vector space is equal to the number of vectors in its basis.
• This is because the basis vectors are linearly independent, and therefore the dimension of the space is equal to the number of basis vectors needed to span the space.
• The dimension of a vector space is a fundamental property that characterizes the space.