Math 303: Section 24

Dr. Janssen

\[ \def\R{{\mathbb R}} \def\P{{\mathbb P}} \def\b{{\mathbf{b}}} \def\x{{\mathbf{x}}} \def\y{{\mathbf{y}}} \def\u{{\mathbf{u}}} \def\v{{\mathbf{v}}} \def\w{{\mathbf{w}}} \def\z{{\mathbf{z}}} \def\M{{\mathcal{M}}} \DeclareMathOperator{\null}{Nul} \DeclareMathOperator{\span}{Span} \DeclareMathOperator{\dim}{dim} \newcommand{\set}[1]{\left\{ {#1} \right\}} \newcommand{\setof}[2]{{\left\{#1\,\colon\,#2\right\}}} \]

Upshot: Every basis for a vector space \(V\) has the same number of vectors in it.

**Definition 1 **A **finite-dimensional** vector space is a vector space that can be spanned by a finite number of vectors. The **dimension** of a non-trivial finite-dimensional vector space is the number of vectors in a basis for \(V\). The dimension of the trivial space is defined to be 0. We denote the dimension of \(V\) by \(\dim(V)\).

Find the dimensions of each of the following spaces. Justify your answers.

- \(\P_1\)
- \(\P_2\)
- \(\P_n\)

- \(\M_{2\times 3}\)
- \(\M_{3\times 4}\)
- \(\M_{k\times n}\)

Let \(W = \setof{(a+b) + (a-b) t + (2a+3b)t^2}{a,b\in\R}\).

- Find a finite set of polynomials in \(W\) that span \(W\).
- Determine if the spanning set you found is linearly independent or dependent. Clearly explain your process.
- What is \(\dim(W)\)? Explain.

Let \(V\) be a finite dimensional vector space of dimension \(n\) and let \(W\) be a subspace of \(V\). Explain why \(W\) cannot have dimension larger than \(\dim(V)\), and if \(W\ne V\), then \(\dim(W) < \dim(V)\).

Typically, we need to verify two properties to confirm that a given set \(S\) of vectors forms a basis for a vector space \(W\):

- \(W = \span(S)\)
- \(S\) is linearly independent.

**But!** If we know \(\dim(W)\) ahead of time, things are a bit simpler.

For all that follows, let \(W\) be a subspace of a vector space \(V\) with \(\dim(W) = k\).

Suppose first that \(S\) is a subset of \(W\) containing \(k\) vectors, and is linearly independent. Let’s show that \(\span(S) = W\).

- Suppose \(S\) does not span \(W\). Explain why this implies that \(W\) contains a set of \(k+1\) linearly independent vectors.
- Explain why the result of the previous part tells us that \(S\) is a basis for \(W\).

Now suppose that \(S\) is a subset of \(W\) with \(k\) vectors that spans \(W\). Let’s show \(S\) is linearly independent.

- If \(S\) is not linearly independent, explain why we can then find a proper subset of \(S\) that is linearly independent and has the same span as \(S\).
- Explain why the result of the previous part tells us that \(S\) is a basis for \(W\).

**Theorem 1 **Let \(W\) be a subspace of dimension \(k\) of a vector space \(V\) and let \(S\) be a subset of \(W\) containing exactly \(k\) vectors.

- If \(S\) is linearly independent, then \(S\) is a basis for \(W\).
- If \(S\) spans \(W\), then \(S\) is a basis for \(W\).