Linear Algebra Coordinate Systems Flashcards

Coordinate Systems and Vector Spaces

• If B is the standard basis for ℝn, then the B-coordinate vector of any vector x in ℝn is equal to x itself, represented as [x]B = x.
• The standard basis is made up of the columns of the n×n identity matrix, supporting the conclusion that [x]B = x_1e_1 + ... + x_ne_n = x.

Coordinate Mapping

• The mapping that relates vectors to their B-coordinates is defined as x↦[x]B, not the other way around, which is a common misconception.

Isomorphic Spaces

• A plane in ℝ3 that passes through the origin is isomorphic to ℝ2, indicating a structural equivalence in their dimensionality.

Uniqueness Representation Theorem

• For any basis B = {b_1,..., b_n} of a vector space V, each vector x in V can be expressed uniquely as a linear combination of the basis vectors, defined as x = c_1b_1 + ... + c_nb_n.

Definition of Isomorphism

• An isomorphism is precisely a one-to-one linear transformation that maps a vector space V onto another vector space W, ensuring both spaces have the same dimension and structure.

Coordinate Vector Representation

• If a vector space V has a basis B consisting of n vectors, then the B-coordinate vector of any vector x in V will reside in ℝn.

Change of Coordinates Matrix Misunderstanding

• The statement that the relationship [x]_B = P_Bx is false; this implies misunderstanding of how the change of coordinate matrix functions in a vector space context.

Isomorphic Relation of P_3 and ℝ3

• The vector space P_3, which consists of polynomials of degree less than or equal to 3, is not isomorphic to ℝ3; instead, it is isomorphic to ℝ4, indicating a dimensional difference.

Basis Characterization of a Set S

• A finite set S in vector space V, where every vector x has a unique representation as a linear combination of elements from S, confirms that S spans the vector space.
• The linear independence of S is demonstrated by showing that the only linear combination that gives the zero vector is when all scalars c_1, ..., c_n are zero, affirming S is a basis for V.