Vector in ℝⁿ

The idea of a vector comes from two sources: physics, where a force or a velocity has magnitude and direction, and data, where a list of measurements is treated as a single object. In both cases what matters is that objects can be combined: added and stretched. Textbooks such as Strang's start in the plane \(\mathbb{R}^2\) and move on to \(\mathbb{R}^3\) and \(\mathbb{R}^n\).

Vectors are defined by components: \((1,2,3)+(4,-1,0)=(5,1,3)\) and \(3(2,-1,4)=(6,-3,12)\). These two operations are all linear algebra needs; the other notions (length, dot product) are extras that depend on geometry. A useful guiding example: the vector \((3,4)\) is the arrow ending at the point \((3,4)\), and \(2(3,4)=(6,8)\) is the same arrow stretched to twice its size.

Typical mistakes: adding vectors of different sizes (\((1,2)+(1,2,3)\) does not exist), confusing the scalar \(0\) with the zero vector \((0,\dots,0)\), and scaling only one component. The norm \(\|x\|=\sqrt{x_1^2+\cdots+x_n^2}\) and the dot product also appear here; they measure lengths and angles.

This concept connects with the earlier ones: a matrix with \(n\) rows and one column is a vector, and a linear system asks to write a vector as a combination of columns. The next step, the vector space, keeps only the properties of addition and scaling and forgets that the objects are lists. Remember that vectors are added componentwise, \(x+y=(x_i+y_i)\), and multiplied by a scalar \(c\) by scaling each component, \(cx=(cx_i)\).