Linear algebraLesson 2 of 5 in the course Vector spaces

Linear Independence: Tests, Bases, and Affine Extensions

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

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Summary

This lesson introduces linear independence and dependence, the central ideas used to determine whether vectors contribute genuinely new directions or information.

For vectors v1,…,vkv_1,\ldots,v_k in a vector space, consider the equation

The vectors are linearly independent if this equation forces every coefficient to be zero: a1=⋯=ak=0a_1=\cdots=a_k=0. This is called the trivial solution.

All flashcards

Linear independence
A list is independent when the only coefficients producing the zero vector are all zero.
Nontrivial relation
A linear combination equal to zero with at least one nonzero coefficient.
Dependence and redundancy
In a dependent finite list, at least one vector is a combination of the others.
Independence of functions
A function combination is independent when vanishing for every input forces all coefficients to be zero.
Relation space
The null space of the column matrix contains all coefficient tuples producing the zero vector.

Quiz questions

  1. If vectors are placed as the columns of a matrix A, when are the columns linearly independent?Answer: When A a = 0 has only the solution a = 0.
  2. For exactly n vectors in R^n, which determinant condition establishes linear independence?Answer: The determinant of the column matrix is nonzero.
  3. What must be true of more than n vectors in an n-dimensional vector space?Answer: They are linearly dependent.
  4. What two properties define a basis for a vector space?Answer: The set is linearly independent and spans the space.
  5. How can affine independence of vectors v_i be tested using ordinary linear independence?Answer: Test whether the augmented vectors (1, v_i) are linearly independent.

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

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