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Section 8.1 Definition of a Vector Space

In this section, we give the formal definitions of a vector space and list some examples.

Definition 8.1. Vector Space.

A set of objects (vectors) \(\{\vec{u}, \vec{v}, \vec{w}, \dots\}\) is said to form a linear vector space over the field of scalars \(\{\lambda, \mu,\dots\}\) (e.g. real numbers or complex numbers) if:

  1. the set is closed, commutative, and associative under (vector) addition;

  2. the set is closed, associative, and distributive under multiplication by a scalar;

  3. there exists a zero vector \(\vec{0}\) that leaves other vectors unchanged under addition;

  4. multiplication by the scalar identity \(1\) leaves the vector unchanged;

  5. every vector \(\vec{v}\) have a corresponding negative vector \(-\vec{v}\) such that \(\vec{v}+(-\vec{v})=\zero\text{;}\)

The trick here is not only to identify the set of objects that are in the vector space, but also what is meant by addition of vectors and scalar multiplication.

Some examples of vector spaces are:

  1. Forces on a point particle that can move in a plane (i.e. arrows in 2-D).

  2. Forces on a point particle that can move in space (i.e. arrows in 3-D). Notice that arrows in 2-D and arrows in 3-D are different vector spaces.

  3. \(m\times n\) matrices for fixed \(m\) and \(n\text{.}\)

  4. Sufficiently smooth functions on the interval \(0\le x\le L\) that go to zero at \(x=0\) and \(x=L\text{,}\) as in quantum particle-in-a-box.

  5. Periodic functions with a fixed period, e.g. Fourier Series.

Activity 8.1. Definition of addition for example vector spaces.

For each of the example vector spaces above, state what is meant by the sum of two vectors.
Answer.

For arrows, addition means the parallelogram rule. For matrices, addition means component by component, which is equivalent to the parallelogram rule if the matrices happen to be columns. For functions, addition is pointwise addition of functions.