Intuition The one core ideaA linear transformation is a machine that turns arrows into arrows without bending straight lines or spacing. Once you agree on a small set of "ruler arrows" (a basis) for the input and output, you only need to record where each ruler arrow lands — that little table of numbers is the matrix, and it lets you predict where every other arrow goes.
This page assumes you have seen none of the notation in the parent note. We build every symbol from the ground up, in an order where each one leans only on the ones before it. Read top to bottom.
A vector is an arrow with a length and a direction. In the flat plane we describe it by two numbers: how far right, and how far up. We write v = ( x , y ) ( 0 , 0 ) ( x , y )
The picture below shows one vector and the two numbers that name it. Look at the burnt-orange arrow: it goes 3 2 ( 3 , 2 )
Intuition Why numbers, not just pictures?A picture of an arrow is nice, but you cannot compute with a picture. Attaching two numbers lets us add, scale, and eventually feed arrows through machines using arithmetic. The whole topic is a translation from pictures of arrows into columns of numbers .
Everything linear is built from exactly two operations.
Definition Addition and scalar multiplication
Add two vectors: put them tip-to-tail; the result is the arrow from the very start to the very end. In numbers, add matching slots: ( a , b ) + ( c , d ) = ( a + c , b + d ) Scale a vector by a number c scalar , just an ordinary number): stretch or shrink the arrow. In numbers, c ⋅ ( a , b ) = ( c a , c b ) c
Definition Linear combinationA linear combination of vectors v 1 , v 2 , … c 1 v 1 + c 2 v 2 + ⋯ c 1 c 2
Why we care: the parent note's central claim — "a basis pins down T every vector is a linear combination of a few chosen arrows . So this idea must come first.
A basis of a space is a smallest set of arrows such that every vector in the space is a linear combination of them in exactly one way. The most familiar basis of the plane is e 1 = ( 1 , 0 ) e 2 = ( 0 , 1 )
The figure shows how the arrow ( 3 , 2 ) built from these ruler arrows: three of e 1 e 2
Intuition Why "exactly one way" is the magic wordIf a vector could be built from the basis in two different ways, its "coordinates" would be ambiguous and no bookkeeping table could work. Uniqueness is what makes the whole matrix idea possible. This uniqueness is exactly what the parent note relies on when it writes "for unique coordinates x i
A basis need not be e 1 , e 2 {( 1 , 1 ) , ( 1 , − 1 )} Coordinate vectors and bases for the full story.
Definition Coordinate vector[ v ] B B = { b 1 , … , b n } v = x 1 b 1 + ⋯ + x n b n x 1 , … , x n coordinates of v B . We stack them in a column and write
[ v ] B = x 1 ⋮ x n . B B
Worked example Same arrow, two coordinate columnsThe arrow ( 3 , 2 ) [ v ] e = ( 2 3 ) B ′ = {( 1 , 1 ) , ( 1 , − 1 )} ( 3 , 2 ) = a ( 1 , 1 ) + b ( 1 , − 1 ) a = 2.5 b = 0.5 [ v ] B ′ = ( 0.5 2.5 ) Same arrow, different numbers — because the ruler changed.
Common mistake Thinking a vectoris its coordinates
Why it feels right: we almost always use the standard ruler, where ( 3 , 2 ) The fix: coordinates only exist relative to a basis . The subscript B
A vector space is any collection of things you can add and scale following the usual rules. Arrows in the plane form one. So do polynomials like 4 + 5 x + 6 x 2
The dimension of a space is the number of arrows in any basis of it — the count of independent "directions." The plane has dimension 2 2 P 2 { 1 , x , x 2 } 3
We write V input space and W output space. Letters n = dim V m = dim W m n
T : V → W T machine (a function) that eats a vector from space V W T : V → W T V W T ( v ) v
T linear if it respects the only two moves from Section 2:
T ( u + v ) = T ( u ) + T ( v ) , T ( c v ) = c T ( v ) . Picture: the grid of straight, evenly-spaced lines stays straight and evenly-spaced after the machine runs — no bending, no curving. Only stretching, rotating, shearing, or flipping.
The figure contrasts a linear machine (a 90°
Intuition Why linearity is the whole gameBecause T T T ( x 1 b 1 + ⋯ + x n b n ) = x 1 T ( b 1 ) + ⋯ + x n T ( b n ) . T n T everything . That finite data is what the matrix will store.
A matrix is a rectangular grid of numbers. We use two bases: B V C W [ T ] C B T B C j [ T ( b j ) ] C j
Definition Matrix times columnA [ v ] B A row of A dot product of that row with the column. The result is a new column — the output coordinates.
Deeper study of these pieces lives in Composition of linear maps , Rank and nullity , Kernel and image , Eigenvalues and diagonalization , and Change of basis . Return to the parent Matrix representation of linear transformations once the symbols below feel automatic.
Add tip to tail and scale
Coordinates and bracket v in B
Vector space V and W dimension
Matrix table T in C from B
Output coords = matrix times input coords
Test yourself. Each line: cover the answer, recall it, then reveal.
What two numbers name a plane vector, and what do they mean? The horizontal step and vertical step from the origin to the arrow's tip, written ( x , y )
What are the only two operations everything linear is built from? Adding vectors (tip-to-tail) and scaling a vector by a number.
What is a linear combination of v 1 , v 2 Any c 1 v 1 + c 2 v 2
What makes a set of arrows a basis? Every vector is a linear combination of them in exactly one way.
Why does "exactly one way" matter? It makes coordinates unambiguous, which is what lets a single matrix work.
What does [ v ] B The column of coordinates of v B
Is a vector the same as its coordinates? No — coordinates depend on which basis you chose; drop the subscript and the meaning is ambiguous.
What is the dimension of a space? The number of arrows in any basis of it.
What does T : V → W T V W
What does it mean for T T ( u + v ) = T ( u ) + T ( v ) T ( c v ) = c T ( v )
Why do the values of T T Every vector is a unique combination of basis vectors, and linearity carries T
What sits in the j [ T ] C B The coordinates of T ( b j ) j C
For T : V → W dim V = n dim W = m m n
State the fundamental relationship. [ T ( v ) ] C = [ T ] C B [ v ] B
