Questions and Answers
If B is the standard basis for ℝn, then the B-coordinate vector of an x in ℝn is x itself.
True
The correspondence [x]B↦x is called the coordinate mapping.
False
In some cases, a plane in ℝ3 can be isomorphic to ℝ2.
True
What is the uniqueness representation theorem?
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What is an isomorphism?
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If x is in V and if B contains n vectors, then the B-coordinate vector of x is in R^n.
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If P_B is the change of coordinates matrix, then [x]_B = P_Bx for x in V.
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The vector spaces P_3 and R3 are isomorphic.
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Show that S is a basis of V if every x in V has a unique representation as a linear combination of elements of S.
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Study Notes
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.
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Test your understanding of coordinate systems and the standard basis in ℝn with these flashcards. Each card presents a statement related to coordinate mapping and the properties of the standard basis. Ideal for students looking to reinforce their knowledge in linear algebra concepts.
