# Linear Algebra

Linear Algebra 1:

Matrix manipulations, lots of higher dimensional thinking, not proof based. I found this course more intuitive than Calculus. Linear dependence, vectors, matrices, determinants, etc.

### Common Symbols in LA:

ℝ = All real numbers (integers, rational, irrational)

ℝ^{n} = n denotes the dimension we are in

ℝ^{2} is the cartesian plane (x,y) aka 2D space

ℝ^{3} is in 3 dimensional space (x,y,z)

#### Let A denote a matrix:

| A | or det(A) = The determinant of Matrix A

I = The identity matrix

E = The elementary matrix

E_{ij} = i denotes the # of rows in the matrix, and j denotes the # of columns

M_{mxn} = All matricies with m rows and n columns

A^{-1} = The inverse matrix of A

A^{T} = The transpose matrix of A

sp(A) or span(A) = The span of A (set of all linear combinations of the vectors/matrices)

dim(A) = The dimension of A

col(A) = Column space of A (span of its column vectors)

row(A) = Row space of A (span of its column vectors)

rank(A) = The rank of A (dimension of row space)

adj(A) = Adjugate matrix (transpose of cofactor matrix)

##### Let A,B denote 2 vectors

→ A = A is a vector

|| A || = The norm of A

A X B = The cross product of A and B (only defined in the 3rd Dimension)

A * B or A . B = The dot product (inner product) of A and B (scalar)

A ⊂ B = A is a subset of B

###### 3rd party resources:

Defining a Plane using a Normal Vector and a Point