Coordinate systems and change of basis are crucial concepts in linear algebra. They allow us to represent vectors uniquely using numbers and switch between different representations. This flexibility is key for solving problems and understanding vector spaces.

Mastering these ideas helps us grasp how vectors can be expressed in various ways. We'll explore how to choose basis vectors, express vectors in terms of a basis, and change between different coordinate systems using transformation matrices.

Coordinate systems in vector spaces

Definition and properties

• A coordinate system in a vector space uniquely identifies and locates vectors using an ordered set of numbers called coordinates
• The number of coordinates needed to specify a vector equals the dimension of the vector space
  • For example, in a 3-dimensional vector space, a vector is specified by 3 coordinates (x, y, z)
• In an n-dimensional vector space, a coordinate system is defined by choosing n linearly independent vectors called basis vectors
• The standard basis for an n-dimensional vector space is the set of n vectors, each with a 1 in one coordinate and 0s in all other coordinates
  • For example, the standard basis for a 2-dimensional vector space is {(1, 0), (0, 1)}

Choice of basis vectors

• The choice of basis vectors is not unique, and different bases can be used to represent the same vector space
• Different bases may be chosen for convenience, simplicity, or to highlight specific properties of the vector space
  • For example, in a 2D plane, a basis could be chosen as {(1, 0), (0, 1)} or {(1, 1), (-1, 1)}, among others
• Changing the basis vectors does not change the underlying vector space, only the representation of vectors within that space

Vectors in terms of a basis

Expressing a vector in a given basis

• To express a vector in terms of a given basis, find the coordinates of the vector with respect to that basis
• The coordinates of a vector v in terms of a basis {b1, b2, ..., bn} are the scalars c1, c2, ..., cn such that v = c1b1 + c2b2 + ... + cnbn
  • For example, if v = (3, 4) and the basis is {(1, 0), (0, 1)}, then v = 3(1, 0) + 4(0, 1)
• The coordinates of a vector can be found by solving a system of linear equations or using the dot product with the dual basis vectors

Dual basis vectors

• The dual basis vectors are a set of vectors {b1*, b2*, ..., bn*} such that bi* · bj = δij (Kronecker delta), where δij = 1 if i = j and 0 otherwise
• The coordinates of a vector v with respect to a basis {b1, b2, ..., bn} can be found using the formula ci = v · bi
  • For example, if v = (3, 4) and the basis is {(1, 0), (0, 1)}, then the dual basis is also {(1, 0), (0, 1)}, and c1 = (3, 4) · (1, 0) = 3, c2 = (3, 4) · (0, 1) = 4
• Dual basis vectors provide a convenient way to find the coordinates of a vector in a given basis

Change of basis

Transformation matrix

• A change of basis expresses a vector in terms of a new set of basis vectors
• To perform a change of basis, find the transformation matrix that relates the old basis to the new basis
• The transformation matrix P from an old basis {b1, b2, ..., bn} to a new basis {b1', b2', ..., bn'} is formed by expressing each new basis vector in terms of the old basis vectors
  • For example, if the old basis is {(1, 0), (0, 1)} and the new basis is {(1, 1), (-1, 1)}, then P = [1 -1; 1 1]
• The columns of the transformation matrix P are the coordinates of the new basis vectors in terms of the old basis vectors

Changing coordinates

• To change the coordinates of a vector v from the old basis to the new basis, multiply the transformation matrix P by the coordinate vector of v in the old basis
  • For example, if v = (3, 4) in the old basis {(1, 0), (0, 1)} and P = [1 -1; 1 1], then [v]new = P[v]old = [7; -1]
• The new coordinate vector [v]new represents the same vector v in the new basis
• Changing the basis allows for the analysis of vectors from different perspectives and can simplify calculations in certain situations

Coordinates in different bases

Linear transformation

• The coordinates of a vector in different bases are related by a linear transformation
• If P is the transformation matrix from basis B to basis B', and [v]B and [v]B' are the coordinate vectors of v in bases B and B' respectively, then [v]B' = P[v]B
  • For example, if v = (3, 4) in basis B, P = [1 -1; 1 1], and B' is the new basis, then [v]B' = P[v]B = [7; -1]
• The inverse of the transformation matrix P, denoted as P^(-1), transforms the coordinates from the new basis back to the old basis, i.e., [v]B = P^(-1)[v]B'

Properties of transformation matrices

• The transformation matrices between two bases are inverses of each other
  • If P is the transformation matrix from basis B to basis B', then P^(-1) is the transformation matrix from basis B' to basis B
• The relationship between the coordinates of a vector in different bases allows for the comparison and analysis of vectors in various coordinate systems
• Understanding the properties of transformation matrices is crucial for performing basis changes and analyzing vectors in different coordinate systems
  • For example, the determinant of a transformation matrix indicates whether the basis change preserves orientation (positive determinant) or reverses orientation (negative determinant)