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Linear Algebra
Step 2 of 22

Adding and scaling

Understand first: vectors

There are exactly two things you may do to vectors. Not "two important things" — two, total. Everything ahead is these two moves in combination.

Adding is walking

(2,1)+(1,3)=(3,4)(2, 1) + (1, 3) = (3, 4). Add the entries.

Geometrically that is a journey. Walk the first vector: 2 right, 1 up. From wherever you land, walk the second: 1 right, 3 up. You finish 3 right and 4 up from where you began.

That is why it is drawn tip to tail. Addition is one displacement followed by another, asking where you ended up.

Order does not matter: east then north lands where north then east does, so u+v=v+uu + v = v + u. Arithmetic and picture agree, which is the bridge from step one doing its job.

Scaling is stretching

2⋅(2,1)=(4,2)2 \cdot (2,1) = (4, 2). Multiply each entry. Same direction, twice as long.

Three cases behave differently and are worth separating:

  • Bigger than 1 stretches the arrow.
  • Between 0 and 1 shrinks it, still on its line.
  • Negative flips it to point the other way: −1⋅(2,1)=(−2,−1)-1 \cdot (2,1) = (-2,-1).

The multiplying number is called a scalar, and the name is honest: its only job is to change scale.

Now notice what never happens. Scaling lengthens, shortens, or reverses — it can never tilt a vector off its own line. Every multiple of (2,1)(2,1) lives on one infinite line through the origin.

File that away. Much later, the vectors that refuse to tilt under a whole transformation will turn out to be the most valuable objects in the subject.

Why so few operations

The restriction is the source of the power. Because only these two moves exist, any function that respects them is severely constrained — and those constrained functions are exactly the useful ones.

Which raises the obvious question: if you may only add and scale, where can you actually get to?