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Linear AlgebraSpan

Lästid: ~15 min

Although there are many operations on columns of real numbers, the fundamental operations in linear algebra are the linear ones: addition of two columns, multiplication of the whole column by a constant, and compositions of those operations. In this section we will introduce some vocabulary to help us reason about linear relationships between vectors.

A linear combination of a list of vectors \mathbf{v}_1, \ldots, \mathbf{v}_k is an expression of the form

\begin{align*}c_1\mathbf{v_1} + c_2\mathbf{v_2} + \cdots + c_k\mathbf{v_k},\end{align*}

where c_1, \ldots, c_k are real numbers. The c's are called the weights of the linear combination.

Suppose that \mathbf{u} = [2,0] and \mathbf{v} = [1,2]. Draw the set of all points (a,b) in \mathbb{R}^2 for which the vector [a,b] can be written as an integer linear combination of \mathbf{u} and \mathbf{v}.

Note: An integer linear combination is a linear combination where the weights are integers.

Solution. A bit of experimentation reveals that the integer linear combinations of these two vectors form a lattice as shown.

The span of a list of vectors is the set of all vectors which can be written as a linear combination of the vectors in the list. We define the span of the list containing no vectors to be the set containing only the zero vector.

Is \mathbf{w} = \begin{bmatrix} 1 \\\ 4 \\\ 0 \end{bmatrix} in the span of \mathbf{u} = \begin{bmatrix} 1 \\\ 0 \\\ 0 \end{bmatrix} and \mathbf{v} = \begin{bmatrix} 1 \\\ 1 \\\ 0 \end{bmatrix}?

Find values \alpha and \beta such that \mathbf{w} = \alpha \mathbf{u} + \beta \mathbf{v}. We have α= and β=

We visualize a set S of vectors in \mathbb{R}^n by associating the vector [v_1, v_2, \ldots, v_n] with the point (v_1,\ldots, v_n)—in other words, we associate each vector with the location of its head when its tail is drawn at the origin. Apply geometric reasoning to solve the following exercises.

The span of two vectors in \mathbb{R}^2

can be any shape
must be either a circle or a line
can be all of \mathbb{R}^2
must be either a line or a point
must be either a line or a point or all of \mathbb{R}^2

The span of three vectors in \mathbb{R}^3

can be any shape
must be a sphere or a line
must be a plane
must be a point, a plane, a line, or all of \mathbb{R}^3
must be a plane, a line, or a point

Solution. The span of a list containing only the zero vector is just the origin. The span of a list containing a single vector \mathbf{v} is a line through the origin, since \alpha \mathbf{v} points in the same direction as \mathbf{v} for any \alpha \in \mathbb{R}. The span of a list containing two non-parallel vectors \mathbf{u} and \mathbf{v} is all of \mathbb{R}^2, since the span consists of the union of all lines which run in the \mathbf{u} direction and pass through any point in the span of \{\mathbf{v}\}. Including more vectors can't increase the span further, so these are the only possibilities. So the correct answer is (e).

The same reasoning implies that the span of a list of vectors in \mathbb{R}^3 must be either the origin, or a line or plane through the origin, or all of \mathbb{R}^3. So the correct answer choice is the fourth one.

Check out the 3Blue1Brown video segment below for some helpful visualizations for spans of vectors in three-dimensional space.

Span is closely related to linear dependence, which we will discuss in the next section.

Bruno Bruno