What is a vector
- Physical Angle: Arrows determined by the direction of the length
- Computer perspective: An ordered list of numbers
When you combine the two, you represent vectors in coordinates, and each vector is an arrow going from the origin
Linear combination, span and basis
Linear combination of two vectors: the sum of the number times the number of two vectors
Convention basis vectors, in the span of the basis vectors each vector can be represented as a linear combination of basis vectors
Take a vector in two dimensions as an example. The basis vectors are (1,0) and (0,1). The span of two vectors that are not zero and not on the same line is the entire plane.
If one vector can increase the span of another, the two vectors are linearly independent, otherwise they are linearly dependent (e.g. on the same line).
Strict definition of a basis: A basis for a vector space is a linearly independent set of vectors that span that space.
Matrix and linear transformation
Linear transformation: the origin remains unchanged and all lines remain straight after transformation, or the grid lines remain parallel and evenly spaced.
As long as we determine the coordinates of the transformed basis vectors, we can determine the transformed positions of any vector in this space.
Linear transformations are represented by matrix multiplication. Suppose the basis vectors are (a, c) and (b, d). For any vector (x, y), the following formula can be used to find the transformed coordinates.
Every time you see a matrix, you can view it as some particular transformation in space
Matrix multiplication: Multiple transformations applied to a variable in turn are equivalent to multiplying the variable by the product of multiple matrices multiplied in turn.