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Given any vector space *V* over a field *F*, we can form its associated projective space *P*(*V*) by using the construction above.

*P*(*V*) = *V* - {0}/~ where ~ is the equivalence relation *u* ~ *v* if *u* = *λ**v* for *u*, *v* ∈ *V* - {0} and *λ* ∈ *F*.

**Examples**

- With this definition
**R***P*^{1}=*P*(**R**^{2}) and**R***P*^{2}=*P*(**R**^{3}) so one needs to be careful about dimensions. - Let
*V*be vector space**C**^{2}over the complex numbers**C**. Then*P*(**C**^{2}) is the complex projective line**C***P*^{1}which (arguing as in the real case) consists of**C**together with a single point at infinity.

Topologically, this is the Riemann sphere*S*^{2}. - Let
*V*be a vector space over a*finite*field*F*. Such a finite field has*p*^{k}elements where*p*is a prime number. Then if*V*has dimension*n*over*F*, we have |*V*- {0}| =*p*^{kn}- 1 and since each line through 0 in this has*p*^{k}- 1 elements on it we get |*P*(*V*)| = (*p*^{kn}- 1)/(*p*^{k}- 1).- For any finite field, with
*n*= 2 we get a projective line with |*F*| + 1 points consisting of*F*together with one extra point.

- If |
*F*| = 2 and*n*= 3 we get a projective plane with 7 points.

The points lie in 3's on 7 lines and each point is the intersection of 3 lines.

Every pair of points determines a line and every pair of lines meet in a unique point.

(One of the lines is represented by a circle on the picture.)

- If |
*F*| = 3 and*n*= 3 we get a projective plane with 13 points.

The points lie in 4's on 13 lines and each point is the intersection of 4 lines.

Once again every pair of points determines a line and every pair of lines meet in a unique point.

Examples like these last two are very interesting combinatorial objects and have many nice properties.

- For any finite field, with

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