Vectors

Vector Addition

• Vectors are written as follows:

  \(\begin{bmatrix} 1 \\ 2 \\ \end{bmatrix}\)

• Addition of two vectors:

  \[ \begin{bmatrix} 1 \\ 2 \\ \end{bmatrix} + \begin{bmatrix} 3 \\ -1 \\ \end{bmatrix} = \begin{bmatrix} 1 + 3 \\ 2 - 1 \\ \end{bmatrix} = \begin{bmatrix} 4 \\ 1 \\ \end{bmatrix} \]

Scalar Multiplication

Scaling vectors: result is a scalar.

Multiply each component by the scalar.

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Linear Combinations

• Scalars applied to each vector which scale each vector.

\[ a \vec{v} + b \vec{w} \]

• The span of \(\vec{v}\) and \(\vec{w}\) is the set of all their linear combinations.

• Individual vectors as arrows; sets of vectors as points.

• For 3D vectors, the span is equal to the 'flat sheet' plane that is the 2D plane created by the set of points included.

• Linear combination of \(\vec{v} , \vec{w} , \vec{u}\) :

\[ a \vec{v} + b \vec{w} + c \vec{u} \]

• If the third vector is not on the same plane as the linear combinations of the first two vectors, then it unlocks access to all 3D vectors.

• Redundant vectors that do not increase the span are linearly dependent. Those which do are linearly independent.

The basis of a vector space is a set of linearly independent vectors that span the full space.

Linear Transformations

• Grid lines remain parallel and evenly-spaced.

• Function that takes vector input, outputs a vector.

Find the coordinates of where a vector lands given the original vector coordinates:

1. Record where the basis vectors, \(\hat{i}\) and \(\hat{j}\) land.
2. Recall that the vector \(\vec{v} = -1 \hat{i} + 2 \hat{j}\) This recalls the prior point where each vector is a scaling of the basis vectors.

A vector with coordinates: $$\vec{v} = \begin{bmatrix} -1 \ 2 \ \end{bmatrix} $$

After a linear transformation, the linear combination after the transformation will be the same as before it.

\[ \begin{equation} \begin{split} \text{ Transformed } \vec{v} & = -1(\text{ Transformed } \hat{i}) + 2(\text{ Transformed } \hat{j}) \\ & = -1 \begin{bmatrix} 1 \\ 2 \\ \end{bmatrix} + 2 \begin{bmatrix} 3 \\ 0 \\ \end{bmatrix} \\ & = \begin{bmatrix} -1(1) + 2(3) \\ -1(-2) + 2(0) \\ \end{bmatrix} \\ & = \begin{bmatrix} 5 \\ 2 \\ \end{bmatrix} \\ \end{split} \end{equation}\]

• Deduce where any vectors land.

\[ \begin{bmatrix} x \\ y \\ \end{bmatrix} \rightarrow x\begin{bmatrix} 1 \\ -2 \\ \end{bmatrix} + y\begin{bmatrix} 3 \\ 0 \\ \end{bmatrix} = \begin{bmatrix} 1x + 3y \\ -2x + 0y \\ \end{bmatrix} \]

Any two-dimensional transformation is completely described by 4 numbers, a 2x2 Matrix.

To transform a vector like this, take the vector coordinates and multiply each by the corresponding side of the transformation matrix:

\[ \begin{bmatrix} 3 & 2 \\ -2 & 1 \\ \end{bmatrix} | \begin{bmatrix} 5 \\ 7 \\ \end{bmatrix}\]

\[ 5\begin{bmatrix} 3 \\ 2 \end{bmatrix} + 7\begin{bmatrix} 2 \\ 1 \\ \end{bmatrix}\]

The first column in the transformation matrix represents the point where the first basis vector lands, the second column represents the point where the second basis vector lands.

General Statement

\[\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = x\begin{bmatrix} a \\ c \end{bmatrix} + y\begin{bmatrix} b \\ d \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}\]