# Linear Combinations, Span, Subspaces, Linear Dependence and Independence

Aug-Nov 2020

## Recap

• Vector space $V$ over a scalar field $F$
• $F$: real field $\mathbb{R}$ or complex field $\mathbb{C}$ in this course
• Operations
• vector addition: $u+v$ for $u,v\in V$
• scalar multiplication: $av$ for $a\in F$ and $v\in V$
• Requirements: addition is Abelian, $1v=v$, distributive properties
• Algebraic/abstract notion of a vector
• Vector is defined through its operations
• Connection to physical nature is not emphasized

## Linear Combinations

• Vectors $v_1,v_2,\ldots\in V$, Scalars $a_1,a_2,\ldots\in F$

Linear combination: $a_1v_1+a_2v_2+\cdots$

• Examples: $V=\mathbb{R}^3$

• $a\ \begin{bmatrix}2\\ -1\\ 1\end{bmatrix}=\begin{bmatrix}2a\\ -a\\ a\end{bmatrix}$

• $2\ \begin{bmatrix}5\\ 3\\ -4\end{bmatrix}+3.5\ \begin{bmatrix}2\\ -1\\ -2\end{bmatrix} = \begin{bmatrix}17\\ 2.5\\ -15\end{bmatrix}$

• $4\ \begin{bmatrix}-2\\ 1\\ 6\end{bmatrix}-9\ \begin{bmatrix}5\\ -1\\ -3\end{bmatrix}+3\ \begin{bmatrix}1\\ 2\\ -2\end{bmatrix} = \begin{bmatrix}-50\\ 19\\ 45\end{bmatrix}$

Linear combinations of a handful of vectors generate infinitely many vectors

Linear combinations are the essence of linear algebra

## Span

Vectors $v_1,v_2,\ldots,v_m\in V$

span$(v_1,v_2,\ldots,v_m)=\{a_1v_1+a_2v_2+\cdots+a_mv_m:a_i\in F\}$

• Examples: $V=\mathbb{R}^2$
1. $v=\begin{bmatrix}1\\2\end{bmatrix}$, span$(v)=\{\begin{bmatrix}a\\2a\end{bmatrix}:a\in\mathbb{R}\}$

• Is $\begin{bmatrix}3\\6\end{bmatrix}$ in the span? Is $\begin{bmatrix}2\\10\end{bmatrix}$ in the span?
2. $v_1=\begin{bmatrix}1\\2\end{bmatrix}$, $v_2=\begin{bmatrix}2\\5\end{bmatrix}$, span$(v_1,v_2)=\{\begin{bmatrix}a+2b\\2a+5b\end{bmatrix}:a,b\in\mathbb{R}\}$

• Is $\begin{bmatrix}3\\6\end{bmatrix}$ in the span? Is $\begin{bmatrix}2\\5\end{bmatrix}$ in the span?

• Is $\begin{bmatrix}2\\10\end{bmatrix}$ in the span?

## Span (continued)

• Examples: $V=\mathbb{R}^3$

1. $v_1=\begin{bmatrix}1\\2\\3\end{bmatrix}$, $v_2=\begin{bmatrix}2\\3\\4\end{bmatrix}$, span$(v_1,v_2)=\{\begin{bmatrix}a+2b\\2a+3b\\3a+4b\end{bmatrix}:a,b\in\mathbb{R}\}$

• Is $\begin{bmatrix}2\\8\\10\end{bmatrix}$ in the span?
2. $v_1=\begin{bmatrix}1\\0\\0\end{bmatrix}$, $v_2=\begin{bmatrix}0\\1\\0\end{bmatrix}$, $v_3=\begin{bmatrix}0\\0\\1\end{bmatrix}$. Is $\begin{bmatrix}x\\y\\z\end{bmatrix}$ in the span?

• Example: $V=\mathbb{R}^{1000}$

• 100 vectors given: $v_1$, $\ldots$, $v_{100}$

• Ask if 101-st vector is in the span?

## Subspaces

$U\subseteq V$ is a subspace if and only if $U$ is closed under vector addition and scalar multiplication.

• Equivalent to saying closed under linear combinations
• $u_1,u_2\in U$ implies $au_1+bu_2\in U$ for any $a,b\in F$.
• Why study subspaces?
• Divide and conquer: understand vector space by understanding subspaces
• Examples
• In $\mathbb{R}^2$ and $\mathbb{R}^3$, lines and planes through origin
• $\{(x,x,y)\in F^3: x,y\in F\}$
• $\{(x,y,z)\in F^3: x+y+z=0\}$

Exercise: span$(v_1,v_2,\ldots,v_m)$ is the smallest subspace containing $v_i$.

## Linear Dependence and Independence

Vectors $v_1,v_2,\ldots,v_m\in V$ are said to be linearly dependent if there exist scalars $a_1,a_2,\ldots,a_m\in F$, not all zero, such that $a_1v_1+a_2v_2+\cdots+a_mv_m=0$.

• Linearly dependent if there is a non-trivial linear combination that can result in the zero vector.

Vectors $v_1,v_2,\ldots,v_m\in V$ are said to be linearly independent if $a_1v_1+a_2v_2+\cdots+a_mv_m=0$ implies $a_i=0$ for $i=1,2,\ldots,m$.

• Linearly independent if only the trivial linear combination results in the zero vector.

## Linear dependence of two vectors

• Examples: In $V=\mathbb{R}^2$, are the following linear dependent or independent?
1. $v_1=\begin{bmatrix}1\\2\end{bmatrix}$, $v_2=\begin{bmatrix}3\\6\end{bmatrix}$

2. $v_1=\begin{bmatrix}1\\2\end{bmatrix}$, $v_2=\begin{bmatrix}2\\5\end{bmatrix}$

• Examples: In $V=\mathbb{R}^3$, are the following linear dependent or independent?
1. $v_1=\begin{bmatrix}1\\2\\5\end{bmatrix}$, $v_2=\begin{bmatrix}3\\6\\15\end{bmatrix}$

2. $v_1=\begin{bmatrix}1\\2\\5\end{bmatrix}$, $v_2=\begin{bmatrix}3\\6\\10\end{bmatrix}$

Exercise: Prove that two vectors are linearly dependent if and only if one is a multiple of the other.

## Linear dependence of 3 vectors

• Examples: In $V=\mathbb{R}^2$, are the following linear dependent or independent?

1. $v_1=\begin{bmatrix}1\\0\end{bmatrix}$, $v_2=\begin{bmatrix}0\\1\end{bmatrix}$, $v_3=\begin{bmatrix}3\\17\end{bmatrix}$

2. $v_1=\begin{bmatrix}1\\2\end{bmatrix}$, $v_2=\begin{bmatrix}2\\5\end{bmatrix}$, $v_3=\begin{bmatrix}3\\17\end{bmatrix}$

• Exercise: Prove that any 3 vectors in $\mathbb{R}^2$ are linearly dependent.

• Examples: In $V=\mathbb{R}^3$, are the following linear dependent or independent?

1. $v_1=\begin{bmatrix}1\\0\\0\end{bmatrix}$, $v_2=\begin{bmatrix}0\\1\\0\end{bmatrix}$, $v_3=\begin{bmatrix}a\\b\\c\end{bmatrix}$

2. $v_1=\begin{bmatrix}1\\2\\5\end{bmatrix}$, $v_2=\begin{bmatrix}3\\6\\7\end{bmatrix}$, $v_3=\begin{bmatrix}a\\b\\c\end{bmatrix}$

## How to establish linear (in)dependence of vectors?

• Example: $V=\mathbb{R}^{1000}$
• 100 vectors given: $v_1$, $\ldots$, $v_{100}$
• Are they linearly independent?
• Equivalent question: Is $v_{100}$ in span$(v_1,\ldots,v_{99})$?
• Special case
• $v_i$ has 1 in $i$-th position and zero in all other positions from 1 to 100
• Are they linearly independent? Yes.
• Result: The question of linear dependence of any set of vectors can be manipulated to make it look like the special case above.
• How? Gaussian elimination