the cedar ledge

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

Table of Contents

    1. How Does Linear Combination Operate?
    2. Reading Vector uu in the Cartesian Coordinate Plane
    3. Examples
      1. Simple Linear Combination Example
  2. How To Cite
  3. References
  4. Discussion:

How Does Linear Combination Operate?

Let vv and ww be any two vectors and let cc and dd represent any real numbers. Multiplying vv and ww by cc and dd respectively yields cvcv and dwdw.

Adding cvcv and dwdw together is the idea of a linear combination.

cv+dw=c[v1v2]+d[w1w2]=[cv1+dw1cv2+dw2] 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 uu in the Cartesian Coordinate Plane

In the case of

c[11]+d[23] 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 [xy]\left[ \begin{array}{c} x \\ y \\ \end{array} \right]

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

c[111]+d[234] 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=[11]v = \begin{bmatrix} 1\\ 1 \end{bmatrix}, w=[23]w = \begin{bmatrix} 2\\ 3 \end{bmatrix}, and c=d=1c = d = 1

cv+dw=1[11]+1[23]=[34] 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.

References

Discussion:

CC BY-SA 4.0 Jacob Zelko. Last modified: May 19, 2024. Website built with Franklin.jl and the Julia programming language.