Chapter 2 - Geometry

Definition. Let $c_1, c_2, ... , c_k$ be scalars, and let $v_1. v_2, ... , v_k$ be vectors in $R^n$. The vector in $R^n$

$c_1v_1+c_2v_2+...+c_kv_k$ is called a linear combination of the vectors $v_1, v_2, ... , v_k$ with weights or coefficients $c_1, c_2, ... , c_k$.

Geometrically, a linear combination is obtained by stretching / shrinking the vectors v1, v2, . . . , vk according to the coefficients, then adding them together using the parallelogram law.

2.2 Vector Equation and Span

Definition. A vector equation is an equation involving a linear combination of vectors with possibly unknown coefficients.

Definition. Let v1, v2, . . . , vk be vectors in Rn. The span of v1, v2, . . . , vk is the collection of all linear combinations of v1, v2, . . . , vk, and is denoted Span{v1, v2, . . . , vk}. In symbols: Span{v1, v2, . . . , vk} = x1 v1 + x2 v2 + · · · + xk vk | x1, x2, . . . , xk in R We also say that Span{v1, v2, . . . , vk} is the subset spanned by or generated by the vectors v1, v2, . . . , vk.

2.3 Matrix Equation

Ax=b

If A is an m × n matrix (m rows, n columns), then Ax makes sense when x has n entries. The product Ax has m entries.

Matrix-vector product is distributive

2.4 Solution Set

Solution set is the span from the parametric vector form

Dimension of the solution set. When there is one free variable in a consistent matrix equation, the solution set is a line, and when there are two free variables, the solution set is a plane, etc. The number of free variables is called the dimension of the solution set.

The solution of an inhomogeneous has a particular solution, or a dilation to the solution set(span)

2.5 Linear Independence

Theorem. A set of vectors {v1, v2, . . . , vk} is linearly dependent if and only if one of the vectors is in the span of the other ones. Any such vector may be removed without affecting the span.

2.6 Subspaces

Definition: A subset of R^n is any collection of points of R^n

Definition: A subspace of R^n is a subset V of R^n satisfying:

1. Non-emptiness: The zero vector in V

2. Closure under addition: If u and v are in V then u+v is also in V

3. Closure under multiplication: If v is in V and c is in R, then cv is also in V

Definition: The column space of A is the span, linearly independent vectors

The null space of A contain all solution, linearly dependent vectors

2.7 Basis

Basis is just the basics of a span, recall that span consists unlimited vectors in R^n, however, it needs a basis to be extended with.

RREF and reduce to parametric form, then find the corresponding linearly independent in the regular form