What is a basis of a vector space?

A linearly independent set that spans the vector space.

Why does a basis give unique coordinates?

Spanning ensures a representation exists; independence ensures no two representations differ.

What is a minimal spanning set?

A spanning set from which removing any vector destroys the spanning property.

What is a maximal linearly independent set?

An independent set to which no vector can be added while preserving independence.

Why is a basis usually ordered when coordinates are used?

The order pairs each coordinate position with a particular basis vector.

Linear algebraLesson 1 of 2 in the course Vector spaces

Vector-Space Bases: Dimension, Coordinates, and Change of Basis

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

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Summary

A basis is a compact set of vectors that can build every vector in a space without redundancy. This lesson develops the two defining properties of a basis and explains why they guarantee unique coordinates.

Let $V$ be a vector space over a field $F$ , such as the real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$ . A subset $B$ of $V$ is a basis when it is both linearly independent and spanning.

All flashcards

What is a basis of a vector space?: A linearly independent set that spans the vector space.
Why does a basis give unique coordinates?: Spanning ensures a representation exists; independence ensures no two representations differ.
What is a minimal spanning set?: A spanning set from which removing any vector destroys the spanning property.
What is a maximal linearly independent set?: An independent set to which no vector can be added while preserving independence.
Why is a basis usually ordered when coordinates are used?: The order pairs each coordinate position with a particular basis vector.

Quiz questions

What does the dimension of a vector space measure?Answer: The number of elements in any basis of the space.
In an $n$ -dimensional vector space, when is a set of exactly $n$ vectors a basis?Answer: Exactly when it is linearly independent, or equivalently, when it spans.
Let the columns of $A$ be the new basis vectors expressed in old-basis coordinates. If $Y$ and $X$ are the new and old coordinate columns, respectively, what relation holds?Answer: $X=AY$
What distinguishes a Hamel basis from an analytic basis that permits convergent series?Answer: Hamel expansions use finitely many terms; analytic expansions may use convergent infinite series.
What is true of modules over rings compared with vector spaces over fields?Answer: A module need not have a basis; a module that has one is called free.

Based on the Wikipedia article Basis (linear algebra), licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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