what is basis
A basis is one of those words that has a simple everyday meaning and also some very specific technical meanings, especially in math and finance.
Everyday meaning
In general English, basis means the foundation or main support for something.
Examples:
- “On a daily basis” = happening every day.
- “On what basis did you decide?” = what reasons or facts your decision is built on.
Think of it as the ground or starting point on which something else stands.
In mathematics (linear algebra)
In linear algebra, a basis of a vector space is a set of vectors that:
- Are linearly independent (none of them can be made from combining the others).
- Span the whole space (every vector in that space can be written as a combination of them).
Example in 2D:
- The two vectors (1,0)(1,0)(1,0) and (0,1)(0,1)(0,1) form a basis of the plane.
- Any point (a,b)(a,b)(a,b) can be written uniquely as a(1,0)+b(0,1)a(1,0)+b(0,1)a(1,0)+b(0,1).
All possible bases of a given vector space have the same number of vectors; that number is called the dimension of the space.
Tiny story-style illustration
Imagine you’re describing any color on your screen using just red, green, and
blue sliders.
Those three “pure” colors act like a basis : by mixing them in different
amounts, you can reach any color on the screen, and there’s exactly one way to
mix them for each color.
In finance and tax
In finance and tax, basis usually means your investment amount in an asset, often its cost, adjusted over time.
- Tax basis is used to calculate gain or loss when you sell something.
- Roughly: Gain = selling price − basis.
Example:
- You bought an asset for 2 dollars (your basis is 2).
- Years later you sell it for 100.
- Your gain is 98, because the tax law treats you as having 2 dollars “already in” the asset and 98 as new gain.
Basis can be adjusted over time (for example, increased by improvements or decreased by depreciation), which is why tax authorities devote full publications to explaining it.
Quick HTML table (different meanings)
| Context | What “basis” means | Example |
|---|---|---|
| Everyday English | Foundation or reason something is built on or done. | [5]“On what basis did you choose this candidate?” | [5]
| Math (linear algebra) | A set of linearly independent vectors that spans a vector space. | [3][7]$$(1,0)$$ and $$(0,1)$$ form a basis of the plane. | [3]
| Tax/finance | Your invested amount in an asset used to compute gain or loss. | [4][10][6]Buy at 50, sell at 80 → gain is 30 if your basis is 50. | [6]
TL;DR
- In normal language : basis = the foundation , reason, or regular pattern something rests on.
- In math : basis = a minimal set of building-block vectors that can generate every vector in a space.
- In finance/tax : basis = your starting value/investment in an asset used to measure profit or loss.
Information gathered from public forums or data available on the internet and portrayed here.