Linear combinations and span

Look very closely at a phone screen (a magnifying glass helps) and you’ll see that each pixel is made of tiny red, green and blue lights. There are no yellow lights and no purple lights. Yellow is red plus green; purple is red plus blue; white is all three at full strength.

In vector language: a color is a list of three brightnesses, and every color the screen can show is

for some amounts $r$, $g$, $b$. Scale some vectors, add them up: that’s a linear combination. A whole rainbow of colors comes from mixing just three.

This lesson asks two questions about any set of arrows. How do I mix them to reach a given target? And which targets can I reach at all?

The real words

• A Linear combination of vectors $v_1, \dots, v_k$ is

  $$c_1 v_1 + c_2 v_2 + \cdots + c_k v_k$$

  for any numbers $c_1, \dots, c_k$, called weights or coefficients.

• The Span of those vectors is the set of all their linear combinations. In the plane:

  • two vectors that aren’t on the same line span the whole plane;
  • two vectors on the same line (one a multiple of the other) span only that line;
  • the zero vector alone spans just the origin.

• There’s a quick test for “same line”: $v$ and $w$ lie on one line exactly when $v_1 w_2 - v_2 w_1 = 0$. (That number will come back as the determinant in lesson 9.)

• The standard basis vectors $\hat{\imath} = (1, 0)$ and $\hat{\jmath} = (0, 1)$ (“i hat”, “j hat”) are the simplest pair that spans the plane. The components of a vector are its weights in this mix:

  $$\begin{bmatrix} 3 \\ 2 \end{bmatrix} = 3\begin{bmatrix} 1 \\ 0 \end{bmatrix} + 2\begin{bmatrix} 0 \\ 1 \end{bmatrix}$$

• In 3D, two vectors (not on one line) span a plane through the origin, and you need a third vector sticking out of that plane to span all of space. That’s why screens need three colors of light, not two.