Essence of Linear Algebra

1. Essence of Linear Algebra

Vector

list of numbers, point, direction and length.

Linear combinations

linear combination of two vector $\vec v$ and $\vec w$: (a and b are Scalar) $a\vec v\ +\ b\vec w$

span

span is the set of all of their linear combinations

Linearly dependent/independent

if ${\vec v}$ can represented with ${a\vec w}$, ${\vec v}$ and ${\vec w}$ are linearly dependent. ${\vec v}=a\vec w$ which means a linear combinations with linearly dependent vector, it doesn’t add any span

Linearly dependent $\vec u$ with $\vec v$ and $\vec w$: for all values of a and b, $\vec u=a\vec v\ +\ b\vec w$ Linearly independent $\vec u$ with $\vec v$ and $\vec w$: for all values of a and b, ${\vec u\ \not=\ a\vec v\ +\ b\vec w}$

basis

the base unit of the coordinate system. ex) we choose to use the [1,0], [0,1] as default basis of xy coordinate

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

Linear transformation

terms

‘Transformation’ is same as a function and means it is moved. input->output ‘Linear’:

1. Lines remain lines (all the horizontal, vertical, diagonal lines should not curved,
   Grid lines remain parallel and evenly spaced)
2. Origin remains fixed

   method

   record the vectors after the basis changes, that is the way that linear transformation affect the vectors.