Linear combination

Take two vectors v and w and two numbers a and b. The vector av + bw is a linear combination of v and w with coefficients a and b: a times v followed by b times w. The coefficients can be any real numbers. A number between 0 and 1 takes part of the arrow, and a negative number turns it round.

With Any chosen, the two legs are thin blue arrows, a·v from O and then b·w from its tip, and the thick blue arrow from O is their sum, ending at P. The small ink dots mark the recipes with integer coefficients. When v and w are not collinear, changing only a moves P parallel to v, and changing only b moves it parallel to w, so a and b are P’s coordinates on the slanted grid of v and w. The pale blue layer is everything the allowed recipes reach.

If v and w are not collinear, every point of the plane is a linear combination of them in exactly one way. A construction shows why. Through P draw the line parallel to w. It meets the line of v at exactly one point, which is a·v for exactly one a, and the rest of the way, from a·v to P, runs along w, so it is b·w. Parallels draws this in yellow.

If v and w lie on one line through O, every combination stays on that line, and the slanted grid collapses into it. Points off the line cannot be reached, and every point on it has endless recipes. With v = (2, 1) and w = −2v, the point (4, 2) is 2v + 0w, but also 0v − 1w and 4v + 1w. If both vectors are zero, only O is reached.

Now make the coefficients add up to 1, with a = 1 − μ and b = μ. Then P − v = μ(w − v), so when v ≠ w, P moves along the line through the two tips. It is at v’s tip when μ = 0, at w’s tip when μ = 1, and in proportion in between. P lies between the tips exactly when both weights are at least 0. Past w’s tip, the weight 1 − μ on v is negative.

When v and w are not collinear, fixing the sum a + b = k instead puts P on the line through kv in the direction of w − v, parallel to the line through the tips. Equal sums draws these lines for integers k. They are evenly spaced, and their distance from O grows in proportion to |k|.

For v and w not collinear, limits on the coefficients give shapes. With a, b ≥ 0 (Wedge) the recipes fill the angle between the rays of v and w. With 0 ≤ a, b ≤ 1 (Parallelogram) they fill the parallelogram on v and w. With a, b ≥ 0 and a + b ≤ 1 (Triangle) they fill the triangle with corners O and the two tips.