Vectors and vector spaces
A vector is a quantity with a direction and a length. We draw it as an arrow and write it as numbers, such as (2, 1). Two things can always be done to vectors: add them (put the arrows tip to tail) and scale them (stretch or shrink by a number).
A vector space is a collection of vectors where adding two of them, or scaling one, always gives another vector that is still in the collection. The flat plane, with all arrows from the origin, is a vector space. So is ordinary 3D space. The zero vector must be inside every vector space.
A plane that does not pass through the origin is not a vector space: scale a vector by 0 and you leave it.
Linear combinations and span
A linear combination of u and w is any vector a·u + b·w, where a and b are numbers. In the 3D, the green arrow is exactly this: first go a steps along u, then b steps along w.
The span of some vectors is the set of all their linear combinations. One non-zero vector spans a line. Two vectors that point in different directions span a plane. Three vectors that do not lie in a common plane span all of 3D space.
Linear independence, basis and dimension
Vectors are linearly independent if none of them can be built from the others. If w = 2u, then w adds nothing, so they are dependent. Dependent vectors waste effort: the span does not grow.
A basis is a set of independent vectors that span the whole space. With a basis, every vector has exactly one set of coordinates. The usual basis of the plane is i = (1, 0) and j = (0, 1), but many other pairs also work.
The number of vectors in a basis is the dimension. A line has dimension 1, the plane 2, our room 3. Any basis of the same space has the same size.
Matrices as linear maps
A linear map sends vectors to vectors and keeps both rules: it sends u + w to (image of u) + (image of w), and it sends a·u to a·(image of u). Straight lines stay straight, the origin stays fixed, and the grid stays evenly spaced (it may tilt or stretch).
A matrix describes a linear map. Its columns are the places where i and j land. For M with columns (1, 0.3) and (0.8, 1), the arrow i goes to (1, 0.3) and j goes to (0.8, 1). Because the map is linear, the point (a, b) goes to a·(1, 0.3) + b·(0.8, 1). Nothing else needs to be remembered.
The determinant tells how the area of the unit square changes. If it is 0, the map squashes the plane onto a line and the span drops: a dimension is lost. Read more about the two-by-two case in Matrices as functions.
Key formulas and definitions
- Linear combination: a·u + b·w
- span{u, w} = all vectors a·u + b·w
- u, w dependent if w = k·u (in the plane)
- dimension = number of vectors in a basis
- Linear map: T(u + w) = T(u) + T(w), T(a·u) = a·T(u)
- M(a, b) = a·(column 1) + b·(column 2)
- 2×2 determinant: det [[p, q], [r, s]] = ps − qr (area factor)
Worked examples
1. Let u = (1, 2) and w = (3, 0). Find 2u + w.
2u = (2, 4). Add w: (2 + 3, 4 + 0) = (5, 4).
2. Write (7, 2) as a·(1, 0) + b·(0, 1).
(7, 2) = 7·(1, 0) + 2·(0, 1), so a = 7 and b = 2.
3. Are (1, 2) and (3, 6) independent? What is their span?
(3, 6) = 3·(1, 2), so one is a multiple of the other. They are dependent. Their span is only the line through (1, 2), dimension 1.
4. Are (2, 1) and (1, 3) a basis of the plane?
Check the determinant: 2×3 − 1×1 = 5, which is not 0. So the arrows are independent. Two independent vectors in the plane span it, so they are a basis.
5. The matrix M has columns (2, 0) and (0, 3). Where does (1, 1) go?
M(1, 1) = 1·(2, 0) + 1·(0, 3) = (2, 3). The map stretches x by 2 and y by 3.
6. A map has columns (1, 2) and (2, 4). What does it do to the whole plane?
The determinant is 1×4 − 2×2 = 0. The second column is 2 times the first, so every image lies on the line through (1, 2). The plane is squashed onto a line: the output has dimension 1.
Common mistakes
- Thinking any two vectors form a basis. They must point in different directions (be independent).
- Forgetting that a vector space must contain the zero vector. A plane that misses the origin is not one.
- Reading the matrix by rows instead of columns when asking where i and j land.
- Believing a bigger set of vectors always has a bigger span. Dependent vectors add nothing.