Chapter 6: Vectors and Span

Chapter 6 — Vectors, Linear Combinations, and Span

The equation $Ax=b$ can be read column by column: $Ax=x_1a_1+\cdots+x_na_n$. Solving a system asks whether $b$ can be built from the columns of $A$.

vectorslinear combinationsspancolumn picture

6.1

Vectors and linear combinations

Vector

A vector in $\mathbb R^n$ is an ordered list of $n$ real numbers. It may represent an arrow, a point, a signal, an image, or a data feature vector.

Linear combination

A linear combination of $v_1,\ldots,v_k$ is

$$c_1v_1+\cdots+c_kv_k.$$

Interactive: compute a linear combination

Columns $v_1,v_2,\ldots$ as a matrix

Coefficients $c$

6.2

Span in $\mathbb R^2$

One nonzero vector spans a line through the origin. Two vectors in different directions span all of $\mathbb R^2$.

Interactive: build $c_1v_1+c_2v_2$

$v_1=(x,y)$

$v_2=(x,y)$

$c_1$ 2

$c_2$ 3

Target $b$

6.3

Membership in a span

To test whether $b\in\operatorname{span}\{v_1,\ldots,v_k\}$, put the vectors as columns of $A$ and solve $Ax=b$.

Interactive: is $b$ in the span of the columns?

Matrix with columns $v_i$

Vector $b$

6.4

$Ax=b$ as a column picture

Row picture

Each row is an equation. In $\mathbb R^2$, equations are lines; solving means finding their intersection.

Column picture

The columns are building blocks. Solving means finding coefficients so their weighted sum equals $b$.

$$\begin{bmatrix}1&1\\2&-1\end{bmatrix}\begin{bmatrix}2\\3\end{bmatrix}=2\begin{bmatrix}1\\2\end{bmatrix}+3\begin{bmatrix}1\\-1\end{bmatrix}=\begin{bmatrix}5\\1\end{bmatrix}.$$

6.5

Chapter summary

Span

$\operatorname{span}\{v_1,\ldots,v_k\}$ is the set of all linear combinations of the vectors.

Consistency

$Ax=b$ is consistent exactly when $b$ lies in the span of the columns of $A$.

6.6

Quick checks

What is the span of one nonzero vector in $\mathbb R^2$?

A line through the origin.

Can two vectors span all of $\mathbb R^3$?

No. At most, two vectors span a line or a plane.

What does $Ax=b$ ask in the column picture?

Whether $b$ is a linear combination of the columns of $A$.