Components

A vector has components. \begin{bmatrix} x \\ x \\ x \end{bmatrix}

A matrix has components. \begin{bmatrix} x & x & x \\ x & x & x \\ x & x & x \end{bmatrix}

A + B = C

At some point we learn that if we want to add two things, A and B, the things must have identical structures (structures are determined by components) and the answer , C, has the same structure. We fill the components of the new structure by adding like components of the two structures that we wanted to add:

  • A = (1,2,3)
  • B = (2,5,6)
  • C = (3,7,9)

Hopefully all the above is sufficient to give you the basic idea.

\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + \begin{bmatrix} 2 \\ 5 \\ 6 \end{bmatrix} = \begin{bmatrix} 3 \\ 7 \\ 9 \end{bmatrix}

Appendix A

Sooner or later someone will tell you that describing a vector with components is only one of several ways to represent a vector. We can illustrate ideas of vectors by drawing arrows on a chalkboard or paper.

Someone might also mention that there is a problem in that when we represent a vector by components, the components are as much about the coordinate system as they are about the vector.

Appendix B

One problem with the word is that if we are talking about a vector, and we show an example below…

v = v^1e_1 + v^2e_2 + v^3e_3

…someone might be using the word component for v^2e_2 rather than v^2.