Linear Combinations

You know how to scale a vector. You know how to add two vectors. Put those two ideas together and you get one of the most important concepts in all of linear algebra: the linear combination.

The idea

Take two vectors $\mathbf{v}_1$ and $\mathbf{v}_2$. Scale each one by some number, then add the results:

The numbers $a$ and $b$ are called coefficients (or weights). By choosing different values for them you get different results. The collection of all possible results is something we'll explore in the next lesson.

A concrete example

Let $\mathbf{v}_1 = [2,\ 0]$ (points right) and $\mathbf{v}_2 = [0,\ 2]$ (points up). Then:

• $1.5\,\mathbf{v}_1 + 1\,\mathbf{v}_2 = [3,\ 2]$
• $0\,\mathbf{v}_1 + 2\,\mathbf{v}_2 = [0,\ 4]$
• $-1\,\mathbf{v}_1 + 0.5\,\mathbf{v}_2 = [-2,\ 1]$

Every point on the grid can be reached this way — just pick the right $a$ and $b$.

Try it

In the interactive, v₁ and v₂ are fixed. Use the sliders to set $a$ and $b$ and land the result on the ★ target. The arrow goes blue → pink → result, showing the two steps of the combination geometrically.

Why this matters

Linear combinations are everywhere in applications:

• Computer graphics: every pixel colour is a weighted sum of red, green, and blue
• Machine learning: a neuron computes a weighted sum of its inputs — that's a linear combination
• Signals: any waveform can be expressed as a weighted sum of sine waves (Fourier series)

In code

import numpy as np

v1 = np.array([2, 0])
v2 = np.array([0, 2])

a, b = 1.5, 1.0
result = a * v1 + b * v2
print(result)   # [3. 2.]