Basis and Dimension

The concepts of basis and dimension provide a way to “measure” the size and complexity of a vector space.

Linear Independence and Spanning

A set of vectors $\{\mathbf{v}_1, \dots, \mathbf{v}_n\}$ is linearly independent if the only solution to $c_1\mathbf{v}_1 + \dots + c_n\mathbf{v}_n = \mathbf{0}$ is $c_i = 0$ for all $i$ . The span of a set is the set of all possible linear combinations.

python

Interactive Lab

Read the code, make a small change, then run it and inspect the output. Runtime setup messages stay outside the terminal so the result remains focused on what the program prints.

Step 1

Inspect the idea

Step 2

Edit the program

Step 3

Run and compare

Definition of a Basis

A basis $B$ for a vector space $V$ is a set of vectors that:

Is linearly independent.
Spans $V$ .

How many vectors are in any basis of R^3?

Dimension

The dimension of a vector space $V$ , denoted $\dim(V)$ , is the number of vectors in any basis for $V$ .

python

Interactive Lab

Read the code, make a small change, then run it and inspect the output. Runtime setup messages stay outside the terminal so the result remains focused on what the program prints.

Step 1

Inspect the idea

Step 2

Edit the program

Step 3

Run and compare

Basis and Dimension

Basis and Dimension

Linear Independence and Spanning

Interactive Lab

Definition of a Basis

How many vectors are in any basis of R^3?

Dimension

Interactive Lab

If a subspace of R^5 has dimension 0, what is in the subspace?