Vectors | Chapter 1, Essence of linear algebra

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0:00Linear algebra is built upon the foundation of vectors. To understand them fully, we must look through three distinct lenses. First, the physicist sees a vector as an arrow in space defined solely by its length and direction; it can be moved anywhere without changing its identity. Second, the computer scientist views a vector simply as an ordered list of numbers, like coordinates for a house's size and price. Third, the mathematician generalizes this to any object that supports sensible addition and scalar multiplication. For our purposes, we will primarily use the geometric view: arrows rooted at the origin of a coordinate system.

2:53In a 2D plane with x and y axes intersecting at the origin, a vector's coordinates tell us how to travel from the origin to the vector's tip. The first number dictates horizontal movement (right is positive, left is negative), and the second dictates vertical movement (up is positive, down is negative). We write these pairs vertically in brackets to distinguish them from points. This logic extends to 3D space by adding a z-axis, where a vector is represented by a triplet of numbers corresponding to movements along the x, y, and z axes respectively.

4:37The two most critical operations are vector addition and scalar multiplication. Adding two vectors geometrically involves placing the tail of the second vector at the head of the first; the sum is the vector drawn from the start to the finish. Numerically, this corresponds to adding their respective components ($x_1+x_2$ and $y_1+y_2$). Scalar multiplication acts as 'scaling.' Multiplying by a number greater than 1 stretches the vector, a fraction shrinks it, and a negative number flips its direction while adjusting its length. Component-wise, you simply multiply each entry by the scalar. More precisely, length is multiplied by |c|: only 0<|c|<1 shortens it, negative c reverses it, and $c=0$ yields the zero vector. Column notation is a convention; bracket orientation alone is not a mathematical distinction between a point and a vector.

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