Euclidean space geometry: vector operations
The combination of these words may sound frightening. In fact, if you want to succeed with matrix algebra, you need to start drawing inspiration from geometry as early as possible.
Sum of vectors
Definition. The set of all -dimensional vectors with is denoted and is called a Euclidean space.
is a plane. The space we live in is Our intuition doesn't work in dimensions higher than 3 but most facts we observe in real life on the plane and in the 3-dimensional space have direct analogs in higher dimensions. Keep in mind that can be called a vector or a point in depending on the context. When we think of it as a vector, we associate with it an arrow that starts at the origin and ends at the point
Careful inspection shows that the sum of two vectors is found using the parallelogram rule in Figure 1. The rule itself comes from physics: if two forces are applied to a point, their resultant force is found by the parallelogram rule. Whatever works in real life is guaranteed to work in Math.
Exercise 1. Let be the unit vector of the axis and the unit vector of the axis. Find the sum Generalization: if runs over the whole axis and runs over the whole axis, what is the set of resulting sums Further generalization: on the plane take two straight lines and that pass through the origin and are not parallel to one another. If runs over and runs over , what is the set of resulting sums
This seemingly innocuous exercise leads to profound ideas, to be considered later. The answer for the last question is that the sums will cover the whole This fact is written like this: Note that I intentionally use words that emphasize movement and geometry: "runs over" and "covers the whole". One of the differences between elementary and higher mathematics is that the former deals with fixed elements and the latter with sets within which there are movement and change.
Multiplication of a vector by a number
If is a number and is a vector, we put (scaling or multiplication by a number). Scaling by a positive number means lengthening it in case and shortening in case . Scaling by a negative number means, additionally, reverting the direction of , see Figure 2.
Exercise 2. Take a nonzero vector and find the set of all products where runs over
The answer is that is the straight line drawn through the vector or, alternatively, the straight line drawn through the origin and point
The expression is called a linear combination of vectors with coefficients To find you first scale and and then add the results.
Exercise 3. Let be two nonparallel vectors on the plane. Describe the set verbally and geometrically.
Verbally, this is the set of all linear combinations that result when run over Geometrically, this will be the whole plane