>[!summary] Span is the generalization of [[Linear Combinations]] and determines the possible combinations of vectors that can cover a plane > We can reduce span if one of the vectors in a vector equations can be a linear combination of another vector. > **Key Equations:** > Generalization of span: $Span \space B = \{ c_1 v_1 + \dots + c_k v_k \space | c_1 \dots c_k \in \mathbb{R} \}$ > Generalization of vector equation: $\vec{x} = c_1v_1+ \dots +c_kv_k, \quad ,c_1\dots c_k \in \mathbb{R}$ > Reducing span is only possible: >$\begin{array}{c} \text{Some vector $\vec{v_i}$ where $1 \leq i \leq k$ can be written as a linear combination of } \\ \vec{v_1}, \dots, \vec{v}_{i-1}, \vec{v}_{i+1},\dots , \vec{v} _ k\\\\ \text{If and only if:} \\ Span\{\vec{v_1}, \dots, \vec{v_k}\} = Span\{\vec{v_1}, \dots, \vec{v}_{i-1}, \vec{v}_{i+1},\dots , \vec{v} _ k\} \end{array}$ >[!info]+ Read Time **⏱ 4 mins** # What is Span Span is the generalization of [[Linear Combinations]] for $\mathbb{R^n}$. It determines how many possible combinations are vectors covers a type of plane. The formal definitions are the following: >[!warning] Assumptions Assume $B = \{ \vec{v_1} \dots \vec{v_k} \}$ is vectors in $\mathbb{R^n}$ (These are not all type of linear combinations of vectors that are scalar multiples of a vectors, rather just all types of vectors in a plane) $Span \space B = \{ c_1 v_1 + \dots + c_k v_k \space | c_1 \dots c_k \in \mathbb{R} \} $ A vector equation from this definition from span can be defined as the following: $\vec{x} = c_1v_1+ \dots +c_kv_k, \quad c_1\dots c_k \in \mathbb{R} $ # Reducing Span When we make a span of descrite vectors, sometimes one or more of those vectors can be scalar multiples of another. >Take for example an example from An Introduction To Linear Algebra For Science and Engineering by Norman, D., & Wolczuk, D. The solution is adapted and is my original interpretation of steps. If we have a span of the following below:$ \text{Span} \left\{ \begin{bmatrix} 3 \\ 1 \\ -3 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ -2 \end{bmatrix} \right\} $ Notice that by definition our vector equation from span is the following: $\vec{x} = c_1\begin{bmatrix} 3 \\ 1 \\ -3 \end{bmatrix} + c_2\begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} + c_3\begin{bmatrix} 3 \\ 0 \\ -2 \end{bmatrix}, \quad c_1,c_2,c_2 \in \mathbb{R} $ Notice that the first two vector equation added together give us the last vector equation, so we can rewrite the vector equation as the following (assume $c_1,c_2,c_2 \in \mathbb{R}$ is always true) $\begin{array}{c} \begin{bmatrix} 3 \\ 1 \\ -3 \end{bmatrix} + \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \\ -2 \end{bmatrix} \\ \\ \vec{x} = c_1\begin{bmatrix} 3 \\ 1 \\ -3 \end{bmatrix} + c_2\begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} + c_3\begin{bmatrix} 3 \\ 0 \\ -2 \end{bmatrix} \\ \vec{x} = c_1\begin{bmatrix} 3 \\ 1 \\ -3 \end{bmatrix} + c_2\begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} + c_3 \left\{ \begin{bmatrix} 3 \\ 1 \\ -3 \end{bmatrix} + \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix}\right\} \\\\ \vec{x} = (c_1 + c_3 )\begin{bmatrix} 3 \\ 1 \\ -3 \end{bmatrix} + (c_2 +c_3)\begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} \\ \text{Cant simplify futher than this so:} \\ \text{Let $c_1 + c_3 = s$} \\ \text{Let $c_2 + c_3 = t$} \\ \\ \vec{x} = s\begin{bmatrix} 3 \\ 1 \\ -3 \end{bmatrix} + t\begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix}, \quad s,t \in \mathbb{R} \end{array}$ This is useful in cases where we don't want scalar multiples of vector equations. ## Generalization From using the example above notice that our vectors' equation before we simplified it is technically the same of the vector equation after simplifying because one vector was a scalar multiple of the other. >[!warning] Assumption Assume $B = \{\vec{v_1}, \dots, \vec{v_k}\}$ which represents all possible vectors are in $\mathbb{R^n}$ in any type of vector equation Some vector where $1 \leq i \leq k$ can be written as a linear combination of $\begin{array}{c} \text{Some vector $\vec{v_i}$ where $1 \leq i \leq k$ can be written as a linear combination of } \\ \vec{v_1}, \dots, \vec{v}_{i-1}, \vec{v}_{i+1},\dots , \vec{v} _ k\\\\ \text{If and only if:} \\ Span\{\vec{v_1}, \dots, \vec{v_k}\} = Span\{\vec{v_1}, \dots, \vec{v}_{i-1}, \vec{v}_{i+1},\dots , \vec{v} _ k\} \end{array}$ --- > 🧠 Enjoy this walkthrough? [Support Math & Matter](https://github.com/rajeevphysics/Obsidan-MathMatter) with a star and help others learn more easily. ---