# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Plane vectors in a fixed basis: componentwise operations, equality and the direction of subtraction](/en/derivations/derivation-mko-vector-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-vector)

In the fixed standard basis let u=(3,−1), v=(−2,4), w=(1,3) and k=−2. Compute u+v, u−v, v−u, ku, 0u and 2u+v componentwise; find the vector x with u+x=v; check k(u+v)=ku+kv; and decide whether u=w and whether u=(−1,3).

Plane vectors in a fixed basis

derivation-mko-vector-01



Concept guide



## [Universal closure of a line through zero](/en/derivations/derivation-mko-vector-subspace-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-vector)

U={(x,2x):x in R} is a subspace of R2, whereas U+(0,1) is not.

Real vector subspace

derivation-mko-vector-subspace-01



Concept guide



## [Prove a basis before reading dimension](/en/derivations/derivation-mko-vector-space-dimension-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-vector)

B=((1,1),(1,-1)) is an ordered basis of R2, so dim R2=2.

Dimension of a finite-dimensional vector space

derivation-mko-vector-space-dimension-01





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