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# Linear Combination

Date: May 13 2020

Summary: A loose overview on linear combination and it geometric relationship

Keywords: ##zettel #linearalgebra #vectors ##mathematics #geometry #archive

# Bibliography

Not Available

### How Does Linear Combination Operate?

Let $v$ and $w$ be any two vectors and let $c$ and $d$ represent any real numbers. Multiplying $v$ and $w$ by $c$ and $d$ respectively yields $cv$ and $dw$.

Adding $cv$ and $dw$ together is the idea of a linear combination.

$cv + dw = c\begin{bmatrix} v_{1}\\ v_{2} \end{bmatrix} + d\begin{bmatrix} w_{1}\\ w_{2} \end{bmatrix} = \begin{bmatrix} cv_{1} + dw_{1}\\ cv_{2} + dw_{2} \end{bmatrix}$

### Reading Vector $u$ in the Cartesian Coordinate Plane

In the case of

$c\left[ \begin{array}{c} 1 \\ 1 \\ \end{array} \right] + d\left[ \begin{array}{c} 2 \\ 3 \\ \end{array} \right]$

The entire xy plane is filled. They produce every $\left[ \begin{array}{c} x \\ y \\ \end{array} \right]$

In three element column vectors, a plane is filled in the xyz space:

$c\left[ \begin{array}{c} 1 \\ 1 \\ 1 \\ \end{array} \right] + d\left[ \begin{array}{c} 2 \\ 3 \\ 4 \\ \end{array} \right]$

### Examples

#### Simple Linear Combination Example

Let $v = \begin{bmatrix} 1\\ 1 \end{bmatrix}$, $w = \begin{bmatrix} 2\\ 3 \end{bmatrix}$, and $c = d = 1$

$cv + dw = 1\begin{bmatrix} 1\\ 1 \end{bmatrix} + 1\begin{bmatrix} 2\\ 3 \end{bmatrix} = \begin{bmatrix} 3\\ 4 \end{bmatrix}$

## How To Cite

Zelko, Jacob. Linear Combination. https://jacobzelko.com/05132020134039-linear-combination. May 13 2020.

## Discussion:

CC BY-SA 4.0 Jacob Zelko. Last modified: May 24, 2023. Website built with Franklin.jl and the Julia programming language.