Vector spaces

Vector spaces#

We will primarily work with vectors in \(\mathbb{R}^n\) and occasionally \(\mathbb{C}^n\).

An element of \(\mathbb{R}^n\) is a column vector

\[\begin{split} x = \begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{pmatrix}, \quad x_i \in \mathbb{R}. \end{split}\]

Vector addition#

Given \(x,y \in \mathbb{R}^n\),

\[\begin{split} \begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{pmatrix} + \begin{pmatrix} y_1\\ y_2\\ \vdots\\ y_n \end{pmatrix} = \begin{pmatrix} x_1+y_1\\ x_2+y_2\\ \vdots\\ x_n+y_n \end{pmatrix}. \end{split}\]

Scalar multiplication#

For \(\alpha \in \mathbb{R}\) and \(x \in \mathbb{R}^n\),

\[\begin{split} \alpha \begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{pmatrix} = \begin{pmatrix} \alpha x_1\\ \alpha x_2\\ \vdots\\ \alpha x_n \end{pmatrix}. \end{split}\]

Vectors in \(\mathbb{C}^n\) are defined the same way, with scalars \(\alpha \in \mathbb{C}\).

Definition: Vector space#

A vector space \(V\) over a field \(\mathbb{F}\) (such as \(\mathbb{R}\) or \(\mathbb{C}\)) is a set equipped with addition \((x,y)\mapsto x+y\) and scalar multiplication \((\alpha,x)\mapsto \alpha x\) satisfying, for all \(x,y,z\in V\) and \(\alpha,\beta\in\mathbb{F}\):

  1. \(x+(y+z)=(x+y)+z\) (Associativity of addition)

  2. \(x+y=y+x\) (Commutativity of addition)

  3. There exists \(0\in V\) with \(x+0=x\) (Additive identity)

  4. For each \(x\) there exists \(-x\) with \(x+(-x)=0\) (Additive inverse)

  5. \(\alpha(\beta x)=(\alpha\beta)x\) (Associativity of scalar multiplication / compatibility with field multiplication)

  6. \(1x=x\) (Multiplicative identity of the field acts as identity on vectors)

  7. \(\alpha(x+y)=\alpha x+\alpha y\) (Left distributivity of scalar over vector addition)

  8. \((\alpha+\beta)x=\alpha x+\beta x\) (Right distributivity of scalar addition over scalar multiplication)

Subspaces#

A subspace \(S \subset \mathbb{R}^n\) is a nonempty set closed under linear combinations. Equivalently, for any \(x,y \in S\) and \(\alpha,\beta \in \mathbb{R}\),

\[ \alpha x + \beta y \in S. \]

Every subspace of \(\mathbb{R}^n\) is itself a vector space under the same operations.

Linear combinations and span#

Given vectors \(x_1,\dots,x_k \in \mathbb{R}^n\), a linear combination has the form

\[ \alpha_1 x_1 + \alpha_2 x_2 + \cdots + \alpha_k x_k, \quad \alpha_1,\dots,\alpha_k \in \mathbb{R}. \]

The span of \(\{x_1,\dots,x_k\}\) is the set of all linear combinations:

\[ \operatorname{span}\{x_1,\dots,x_k\} = \left\{ \sum_{i=1}^k \alpha_i x_i \;:\; \alpha_1,\dots,\alpha_k \in \mathbb{R} \right\}. \]

The span is always a subspace of \(\mathbb{R}^n\).

Linear independence#

Vectors \(x_1,\dots,x_k\) are linearly independent if the only solution to the homogeneous combination equaling zero is the trivial one:

\[ \sum_{i=1}^k \alpha_i x_i = 0 \quad \Rightarrow \quad \alpha_1=\cdots=\alpha_k=0. \]

If there exists a nontrivial choice of coefficients yielding zero, the vectors are linearly dependent.

Bases and dimension#

A set of vectors \(x_1,\dots,x_k\) is a basis for a subspace \(S\) if:

  1. \(x_1,\dots,x_k\) are linearly independent, and

  2. \(\operatorname{span}\{x_1,\dots,x_k\} = S\).

If \(x_1,\dots,x_k\) form a basis for \(S\), then \(k\) is the dimension of \(S\), written

\[ \dim(S) = k. \]

While a subspace can have many different bases, each basis has the same number of vectors. Hence dimension is well defined.

Example (a plane in \(\mathbb{R}^3\)). Let

\[\begin{split} a_1=\begin{pmatrix}1\\0\\0\end{pmatrix},\quad a_2=\begin{pmatrix}1\\1\\0\end{pmatrix}. \end{split}\]

Then \(a_1\) and \(a_2\) are linearly independent, and

\[\begin{split} \operatorname{span}\{a_1,a_2\} = \left\{ \begin{pmatrix} \alpha\\ \beta\\ 0 \end{pmatrix} :\; \alpha,\beta\in\mathbb{R} \right\}, \end{split}\]

which is the \(x_1\)-\(x_2\) plane in \(\mathbb{R}^3\). Therefore \(\dim(\operatorname{span}\{a_1,a_2\})=2\).

Direct sum#

If \(U\) and \(V\) are subspaces, then \(U+V\) is a subspace. We say that \(W=U \oplus V\) is the direct sum of \(U\) and \(V\) if \(U \cap V = \{0\}\). The direct sum means that if a vector is decomposed into its \(U\) and \(V\) components, this decomposition is unique.

Example: verify that if \(x_1\), …, \(x_k\) are linearly independent, then the space \(S\) spanned by these vectors is the direct sum of their individual spans:

\[ S = \text{span}\{x_1\} \oplus \cdots \oplus \text{span}\{x_k\} \]
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