Linear alg

2026

Linear combination

Vector form of a line

x=td+p, where l={x:x=td+p for some t exist in R}

Vector form of a plane

vector x = td_1 + sd_2 + p. That is, P={x:x=td₁+sd₂+p for some t,s exist in R}

Span

Linear dependence (geometric)

Linear dependence (algebraically)

Homogenous system of linear equations

Orthogonal

Normal vector

Normal form of a line

Subspace

contains a 0 vector

closed under scalar multiplication

closed under vector addition

Basis

Subspace-Span theorem

Image of a set

Image of a set X under transformation L is the set of all outputs of L when the inputs come from X

Linear transformation

Transformation T is linear of it distributes over addition and scalar multiplication

Rank of a linear transformation

Rank is used to measure compressibility. Rank 0 means sends everything to vector 0, 1 to a line, etc.

Null space/kernel

The kernel is a subspace and describes how many columns are linearly dependent in a set of vectors

Transpose

Switching the columns and rows of a matrix

One-to-one/injective

Every column has a pivot

Onto/surjective

Every row has a pivot

Bijective

All columns and rows have pivots

Inverse matrix

Set equal to identity matrix and solve

Elementary matrix

Change of basis matrix

Determinant

det(A)=ad-cb

Eigenvector

Eigenvalue

fill out

Characteristic polynomial

Diagonalizable

A matrix is diagonalizable if it is similar to a diagonal matrix

Consistent

Has at least one solution

Inconsistent

If there is no solution

Equivalent systems

2 equations or systems of equations are equivalent if they have the same solutions

RREF

1. The first non-entry zero in every row is a 1

2. above and below each leading one are zeros

   1. Leading ones form a staircase pattern to the right and below

Diagonal

If all numbers above and below the diagonal are 0, then the determinant is the product of all the things in the diagonal

Upper and lower triangle

The top/bottom of the lower/upper triagnel include the diagonal (respectively)

Gauss-Jordan theorem

Every system of linear equations can be transformed by row operations into an equivalent system in rref, and the rref is unique

Rouché-Capelli theorem

Talks about number of solutions

1 solution if…

no solution if…

infinite solution if…

Finding bases

The nonzero rows of the rref of a matrix form a basis for its row space

Rank-nullity theorem

For an m*n matrix A,

rank(A)+nullity(A)=n

diagonalization theorem

an n*n matrix A is diagonalizable if A has n linearly independent eigenvectors

A=PDP^-1

Gram-Schmidt process

Given a linearly independent set, the process produces orthogonal basis for the same subspace