Linear algebraLesson 1 of 3 in the course Vector spaces

Vector Spaces: Bases, Maps, Subspaces, Quotients, and Extensions

5 flashcards · 5 quiz questions · 5 match pairs · 3 fill-in sentences · English

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Summary

A vector space is a nonempty set VV together with vector addition and scalar multiplication. The elements of VV are vectors; scalars come from a field FF, such as the real numbers R or complex numbers C. Addition combines two vectors, while scalar multiplication scales a vector by a number.

The operations must satisfy the vector-space axioms. Among them, addition is associative and commutative, there is a zero vector, every vector has an additive inverse, and scalar multiplication interacts consistently with addition and multiplication in the field. These requirements ensure that calculations with vectors are coherent.

All flashcards

Vector space
A set with vector addition and scalar multiplication satisfying the axioms over a field.
Span
The set of all finite linear combinations of vectors in a given set.
Basis
A linearly independent set that spans its vector space.
Isomorphism
A bijective linear map; it shows that two vector spaces have the same linear structure.
Tensor product
A vector space that encodes bilinear relationships through a universal factorization property.

Quiz questions

  1. Which condition makes a map f: V → W linear?Answer: It preserves vector addition and scalar multiplication.
  2. For a square matrix A over a field, when is its associated linear map an isomorphism?Answer: When the determinant of A is nonzero.
  3. For a finite-dimensional linear map represented by A, what condition makes λ an eigenvalue?Answer: The matrix A − λI is not invertible.
  4. If Ax = b has a particular solution x₀, what describes all its solutions?Answer: The set x₀ + ker(A).
  5. What does the first isomorphism theorem state for a linear map f: V → W?Answer: The quotient V/ker(f) is isomorphic to im(f).

Based on the Wikipedia article Vector space, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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