Bases
Definition: A basis of a vector space (V) is a set of vectors ({\mathbf{v}_1, \ldots, \mathbf{v}_k}) that is:
- Linearly independent, and
- Spans (V) (i.e., (\text{span}{\mathbf{v}_1, \ldots, \mathbf{v}_k} = V))
Equivalently, a basis is a minimal spanning set or a maximal linearly independent set.
Standard basis of (\mathbb{R}^n):
[\mathbf{e}_1 = (1,0,\ldots,0), \quad \mathbf{e}_2 = (0,1,\ldots,0), \quad \ldots, \quad \mathbf{e}_n = (0,0,\ldots,1)]
Example: In (\mathbb{R}^2), the set ({(1,0), (0,1)}) is the standard basis. But ({(1,1), (1,-1)}) is also a basis.
Key theorem: All bases of a given vector space have the same number of elements. This number is called the dimension of the space.
Coordinates: Once a basis is chosen, every vector (\mathbf{v} \in V) can be written uniquely as (\mathbf{v} = c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k). The coefficients ((c_1, \ldots, c_k)) are the coordinates of (\mathbf{v}) with respect to that basis.
Parents
Children
- Definition of basis of a vector space (or subspace)
- Definition of the standard/natural basis of R^n (or C^n)
- The standard/natural basis of R^n (or C^n) is a basis.
- Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n)
- Definition of the standard basis of the polynomials of degree at most n
- Definition of the standard basis of the m by n matrices
- Definition of coordinates relative to a given basis
- A set of nonzero vectors contains (as a subset) a basis for its span.
- The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.
- Each vector can be written uniquely as a linear combination of vectors from a given basis.
- A set is a basis if each vector can be written uniquely as a linear combination.
- Coordinates