Vector Spaces, Basis, and Dimension A vector space is a set of vectors (elements) that can be added together and multiplied by scalars (numbers). The set...
A vector space is a set of vectors (elements) that can be added together and multiplied by scalars (numbers). The set of all vectors in a vector space is denoted by the letter V.
A basis for a vector space V is a set of linearly independent vectors that span the entire space. A basis for V is a set of vectors that can be used to represent any vector in V.
The dimension of a vector space V is the number of linearly independent vectors in the basis of V. The dimension of a vector space V tells us how many independent directions are needed to parameterize any vector in V.
Examples:
R³ is a vector space with dimension 3. It is spanned by the vectors i, j, and k.
ℝ² is a vector space with dimension 2. It is spanned by the vectors i and j.
ℝ³ is a vector space with dimension 3. It is spanned by the vectors i, j, and k, as well as the vector e₁ = (1, 0, 0) and e₂ = (0, 1, 0).
By understanding vector spaces, basis, and dimension, we can perform various linear transformations on vectors and analyze the behavior of linear spaces