What is the dimension of a vector space over a specified field?

The cardinality of any basis of the space over that field.

Why is dimension independent of the choice of basis?

All bases of the same vector space have the same cardinality.

What is the dimension of $\mathbb{C}$ over $\mathbb{R}$?

It is $2$, with basis $\{1,i\}$.

How do dimensions compare for a subspace of a finite-dimensional space?

The subspace has no greater dimension; equal dimensions imply the subspace is the whole space.

How are dimensions related across a field extension $F/K$?

For an $F$-vector space $V$, $\dim_K(V)=\dim_K(F)\dim_F(V)$.

Linear algebraLesson 3 of 4 in the course Vector spaces

Vector-Space Dimension: Bases, Subspaces, and Scalar Fields

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

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Summary

Dimension is a way to measure how many independent directions a vector space has. It is defined by counting the vectors in a basis, so understanding bases is essential to understanding dimension.

A basis of a vector space is a set of vectors that is both linearly independent and spans the whole space. The dimension of a vector space $V$ over a field $F$ is the cardinality—the number of vectors—of any basis of $V$ over $F$ . It is also called the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension.

All flashcards

What is the dimension of a vector space over a specified field?: The cardinality of any basis of the space over that field.
Why is dimension independent of the choice of basis?: All bases of the same vector space have the same cardinality.
What is the dimension of $\mathbb{C}$ over $\mathbb{R}$ ?: It is $2$ , with basis $\{1,i\}$ .
How do dimensions compare for a subspace of a finite-dimensional space?: The subspace has no greater dimension; equal dimensions imply the subspace is the whole space.
How are dimensions related across a field extension $F/K$ ?: For an $F$ -vector space $V$ , $\dim_K(V)=\dim_K(F)\dim_F(V)$ .

Quiz questions

If $W$ is a subspace of a finite-dimensional vector space $V$ , which dimension inequality must hold?Answer: $\dim(W)\leq\dim(V)$
For a finite-dimensional vector space $V$ , what follows if a subspace $W$ has the same dimension as $V$ ?Answer: $W=V$
When are two finite-dimensional vector spaces over the same field isomorphic?Answer: When they have the same dimension
If a finite-dimensional vector space $V$ has dimension $n$ over a finite field $F$ , how many elements does $V$ have?Answer: $|F|^n$
For a finite-dimensional real vector space $V$ , what is the trace of its identity operator?Answer: $\dim(V)$

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

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