Linear Algebra

Subspace

A smaller, self-contained math space that exists perfectly flat within a larger one

Basis

The "Goldilocks" set of building blocks for a subspace. It has exactly enough vectors to span the space, but no extra, redundant vectors (they are linearly independent).

Dimension

The number of building blocks in your basis for example: If basis requre two factors, the dimension is 2.

Matrix

A rectangular grid of numbers

Vector (aka Column vector)

A matrix that only has one single column

Identity Matrix

The matrix equivalent of the number 1. If you multiply another matrix by it nothing changes. For example, the image provided is an identity matrix of a 3×3

Inverse Matrix

The matrix equivalent of division. If you multiply a matrix by its inverse, they cancel each other out and leave you with the Identity Matrix. Provided is the inverse of a 2×2, 3×3+ more are different

Transpose

Flipping a matrix over its diagonal. Row become columns and columns become rows

Gaussian Elimination (Row Reduction)

The algorithm or set of steps you use to simplify a matrix.

Row Echelon Form (REF)

A matrix that has been simplified in a staircase pattern where numbers below the staircase are all zero. it dosent have to be 1s, can be any number as long as its stair case.

Reduced Row Echelon Form

the goat, fuly simplified version of a matrix. the staircase is made of entire of 1s and 0s. staircase are 1s and everything else are zero

Pivot

the first non zero number you hit when readinga row left to right. In RREF pivots are 1.

Linear Combination

Mixing and matching vectors. it is the result you get when you multiply a set of vectors by constants (scalars)and add them all together

Span

The set of all possible linear combinations you can make from a specific group of vectors. If you have two vectors pointing in different directions, their span is usually a flat 2D plane.

Linear Independence

Has to to fit the RREF staircase thing.

Linear Dependence

basically if one row is all 0s, dosent fit the RREF stair case thing.

Null space

The collection of all the specific input vectors that get completely crushed to zero when you multiply by the matrix

Column Space

The geometric space spanned by the columns of a matrix