A system of vectors:v1→,...,vn→∈RmAnd a system of scalars:c1,...,cn∈RCan combine to create one unique vector:c1v1→+...+cnvn→\text{A system of vectors:} \\ \overrightarrow{v_1}, ..., \overrightarrow{v_n} \in \mathbb{R}^m \\ \text{And a system of scalars:} \\ c_1, ..., c_n \in \mathbb{R} \\ \text{Can combine to create one unique vector:} \\ c_1\overrightarrow{v_1} + ... + c_n\overrightarrow{v_n} Example:2[−101]+1[010]−3[011]=[−202]+[010][0−3−3]=[−2−2−1]\text{Example:}\\ 2\begin{bmatrix}-1 \\ 0 \\ 1\end{bmatrix} + 1\begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix} - 3\begin{bmatrix}0 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix}-2 \\ 0 \\ 2\end{bmatrix} + \begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix} \begin{bmatrix}0 \\ -3 \\ -3\end{bmatrix} = \begin{bmatrix}-2 \\ -2 \\ -1\end{bmatrix}

Vector Equations

What if, instead of specific numbers, we replaced the values of vectors/scalars with variables?

Variables are denoted as such: x1,x2,...,xnVectors are denoted as such: v1→,v2→,...,vn→,b→∈Rm where b represents constants.The corresponding vector equation would be: x1v1→+x2v2→+...+xnvn→=b→\text{Variables are denoted as such: } x_1, x_2, ..., x_n \\ \text{Vectors are denoted as such: } \overrightarrow{v_1}, \overrightarrow{v_2}, ..., \overrightarrow{v_n}, \overrightarrow{b} \in \mathbb{R}^m \text{ where b represents constants.}\\ \text{The corresponding vector equation would be: } x_1\overrightarrow{v_1} + x_2\overrightarrow{v_2} + ... + x_n\overrightarrow{v_n} = \overrightarrow{b} Example: x[−101]+y[010]+z[011]=[−2−2−1]has a solution of x = 2, y = 1, z = -3 as shown above.We can also rewrite this as [−xy+zx+z]=[−2−2−1] using vector addition.\text{Example: } \\ x\begin{bmatrix}-1 \\ 0 \\ 1\end{bmatrix} + y\begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix} + z\begin{bmatrix}0 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix}-2 \\ -2 \\ -1\end{bmatrix} \\ \text{has a solution of x = 2, y = 1, z = -3 as shown above.} \\ \text{We can also rewrite this as } \begin{bmatrix}-x\\ y + z\\ x + z\end{bmatrix} = \begin{bmatrix}-2\\-2\\-1\end{bmatrix} \text{ using vector addition.}

Solving Vector Equations

We can rewrite a vector equation into an augmented matrix where each of the columns are represented by the vectors.v1→,v2→,...,vn→,b→∈Rm(v1→v2→...vn→b→)\text{We can rewrite a vector equation into an augmented matrix where each of the columns are represented by the vectors.} \\ \overrightarrow{v_1}, \overrightarrow{v_2}, ..., \overrightarrow{v_n}, \overrightarrow{b} \in \mathbb{R}^m \\ \left( \begin{array}{cccc|c} \overrightarrow{v_1} & \overrightarrow{v_2} & ... & \overrightarrow{v_n} & \overrightarrow{b} \end{array} \right) \\

For example,

x[−101]+y[010]+z[011]=[−2−2−1]can be converted into(−100−2011−2101−1)→(101−1011−2−100−2)→(101−1011−2001−3)→(10020101001−3)x\begin{bmatrix}-1 \\ 0 \\ 1\end{bmatrix} + y\begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix} + z\begin{bmatrix}0 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix}-2 \\ -2 \\ -1\end{bmatrix} \\ \text{can be converted into} \\ \left( \begin{array}{ccc|c} -1 & 0 & 0 & -2 \\ 0 & 1 & 1 & -2 \\ 1 & 0 & 1 & -1 \end{array} \right) \rightarrow \left( \begin{array}{ccc|c} 1 & 0 & 1 & -1 \\ 0 & 1 & 1 & -2 \\ -1 & 0 & 0 & -2 \\ \end{array} \right) \rightarrow \left( \begin{array}{ccc|c} 1 & 0 & 1 & -1 \\ 0 & 1 & 1 & -2 \\ 0 & 0 & 1 & -3 \\ \end{array} \right) \rightarrow \left( \begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -3 \\ \end{array} \right)

The Span of a System of Linear Equations

Let some vectors be defined as v1→,v2→,...,vn→∈RmSpan{v1→,v2→,...,vn→}=The set of all combinations in the form c1v1→+...+cnvn→=The set of all b→ such that c1v1→+...+cnvn→=b→ is consistent (has a solution).\text{Let some vectors be defined as } \overrightarrow{v_1}, \overrightarrow{v_2}, ..., \overrightarrow{v_n} \in \mathbb{R}^m \\ Span\{\overrightarrow{v_1}, \overrightarrow{v_2}, ..., \overrightarrow{v_n}\} = \text{The set of all combinations in the form } c_1\overrightarrow{v_1} + ... + c_n\overrightarrow{v_n} = \text{The set of all } \overrightarrow{b} \text{ such that } c_1\overrightarrow{v_1} + ... + c_n\overrightarrow{v_n} = \overrightarrow{b} \text{ is consistent (has a solution).} Examples: This span represents a single line: x = 1Span{[10]}⊂R2=Set of t[10]=[t0]This span represents a single line: x = y Span{[11]}⊂R2=Set of t[11]This span represents the entire x, y planeSpan{[10],[11]}Proof:Let any point on the x, y plane be represented by the vector (xy)(xy)=(x0)+(0y)=x(10)+y(01)=x(10)+y[(11)−(10)]=(x−y)(10)+y(11)Therefore, any point (x,y) can be rewritten as a combination of vectors defined by the span.Intuitively, this proof shows that you can "adjust x" according to your y value to get any point on the plane.\text{Examples: }\\ \text{This span represents a single line: x = 1} \\ Span\{\begin{bmatrix}1\\0\end{bmatrix}\} \subset \mathbb{R}^2 = \text{Set of } t\begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}t\\0\end{bmatrix}\\ \text{This span represents a single line: x = y } \\ Span\{\begin{bmatrix}1\\1\end{bmatrix}\} \subset \mathbb{R}^2 = \text{Set of } t\begin{bmatrix}1\\1\end{bmatrix}\\ \text{This span represents the entire x, y plane} \\ Span\{\begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}1\\1\end{bmatrix}\} \\ \text{Proof:} \\ \text{Let any point on the x, y plane be represented by the vector } \begin{pmatrix}x \\ y\end{pmatrix} \\ \begin{pmatrix}x \\ y\end{pmatrix} \\ = \begin{pmatrix}x \\ 0\end{pmatrix} + \begin{pmatrix}0 \\ y\end{pmatrix} \\ = x\begin{pmatrix}1 \\ 0\end{pmatrix} + y\begin{pmatrix}0 \\ 1\end{pmatrix} \\ = x\begin{pmatrix}1 \\ 0\end{pmatrix} + y[\begin{pmatrix}1 \\ 1\end{pmatrix} - \begin{pmatrix}1 \\ 0\end{pmatrix}] \\ = (x - y)\begin{pmatrix}1 \\ 0\end{pmatrix} + y\begin{pmatrix}1 \\ 1\end{pmatrix}\\ \text{Therefore, any point (x,y) can be rewritten as a combination of vectors defined by the span.} \\ \text{Intuitively, this proof shows that you can "adjust x" according to your y value to get any point on the plane.}

The span of n vectors does NOT have to necessarily span n dimensions; if there is redundant/not enough information, the span can cover fewer dimensions. Also, spans might not be unique; muiltple combinations of vectors can have the same span.

Two vectors that inhabit the same line are said to be colinear. That is, if two vectors are a scalar multiple of one another, then they are colinear.

Span{v1→,v2→} will either be a line if colinear or a plane if not.Span\{\overrightarrow{v_1}, \overrightarrow{v_2}\} \text{ will either be a line if colinear or a plane if not.}