Two arrows in the plane either point along one line or they don’t. Everything in this book is a way of making that precise and then asking what survives when the plane is replaced by something larger.
The object. An ordered pair of arrows (v1, v2) in the plane, and a target point w that the pair is asked to reach. Everything below is about when it can, and how.
A list of arrows is redundant when one of them can be built from the others by stretching and adding. A list has reach in proportion to what it can build. The two ideas pull against each other: add an arrow that can already be built and the reach stays the same; add one that cannot and the reach grows.
Over a field F (for now, the real numbers), vectors v1, …, vk are linearly independent when the only way to write the zero vector as c1v1 + … + ckvk is with every ci = 0. Their span is the set of all such combinations.
The definition is about zero, not about building one vector from the others, and the two are equivalent once you divide by a nonzero coefficient. Dividing needs a field, which is why Chapter 2 spends time on what a field is.
Drag the teal and gold dots above. When the two arrows point along one line, the parallelogram they span is flat and the shading disappears. Away from that, the parallelogram has area, and any point of the plane can be reached. The determinant on the panel measures that area, with a sign.
If the determinant is not zero. Suppose c1v1 + c2v2 = 0, that is, c1a + c2b = 0 and c1c + c2d = 0. Multiply the first by d and the second by b and subtract: c1(ad − bc) = 0. Multiply the second by a and the first by c and subtract: c2(ad − bc) = 0. With ad − bc ≠ 0 both ci are zero.
If the determinant is zero. Then (c1, c2) = (d, −c) gives a combination equal to zero in both components, and so does (b, −a). If c and d are not both zero the first is nonzero; otherwise, if a and b are not both zero, the second is; and if all four entries are zero, (1, 0) works. In every case a nonzero combination vanishes, so the pair is dependent. □
The plate shows the determinant as signed area, but nothing in this chapter proves that; the proof above uses only the two component equations. The trace a + d is not the criterion (Exercise 4). The determinant is defined for larger matrices in Chapter 5, where this proposition will reappear as a special case.
If v1 and v2 are independent, they work like a pair of directions for walking. To reach w, walk some distance c1 along v1, then c2 along v2. Move the arrows and the white dot stays put, but the walking instructions change. The point is one thing; its coordinates are how it is described relative to a choice of directions.
Existence. Substitute. The first component of c1v1 + c2v2 is c1a + c2b = [(w1d − bw2)a + (aw2 − cw1)b]/D = w1(ad − bc)/D = w1. The second is c1c + c2d = [(w1d − bw2)c + (aw2 − cw1)d]/D = w2(ad − bc)/D = w2.
Uniqueness. If two pairs both work, their difference is a combination of v1 and v2 equal to zero. By Proposition 1 (D ≠ 0) the difference is (0, 0). □
The readout “check” on the plate recombines the coordinates and should return w. The link to Book XXI is a pointer, not an identity: a change of basis is the operation under which “the same system” must be judged there, and this chapter says nothing more about those systems.