Linear algebra

Consistent

A system with at least one solution

Inconsistent

A system with no solution

Augmented Matrix

A matrix representing a system, including the constants column

Pivot Position

A position in a matrix that corresponds to a leading 1 in RREF

Free Variable

A variable not associated with a pivot column

RREF (Reduced Row Echelon Form)

A matrix where each pivot is 1, above and below each pivot is 0, and pivots move rightward

Matrix Multiplication

Combine rows of the first matrix with columns of the second

Identity Matrix

A square matrix with 1s on the diagonal, AI=IA=A

Transpose

Flip rows and columns

Symmetric Matrix

A=A^T

Skew-Symmetric Matrix

A^T=−A

Inverse Matrix

Satisfies AA−1=A−1A=IAA^{-1} = A^{-1}A = I

Determinant

A scalar value that determines invertibility and volume scaling

Determinant

det(A) = 0 → matrix is singular (not invertible)

Determinant properties

• Row swap → changes sign

• Scaling a row → scales determinant

• Triangular matrix → product of diagonal entries

Span

Set of all linear combinations of given vectors

Linearly Independent

No vector in the set is a combination of the others

Linearly Dependent

At least one vector is a combination of others

Subspace

• A subset of Rn that:

  1. Contains the zero vector

  2. Is closed under vector addition

  3. Is closed under scalar multiplication

Basis

A linearly independent set that spans the space

Dimension

Number of vectors in a basis for a space

Rank

Number of pivot columns (dimension of column space)

Nullity

Number of free variables (dimension of null space)

Column Space (Im A)

The span of the columns

Null Space (Ker A)

The set of solutions to Ax=0

Linear Transformation

• A function T:Rn→RmT: is linear if:

• T(u+v)=T(u)+T(v)

• T(cu)=cT(u)

Kernel

Set of vectors mapped to 0

Image

Output vectors (column space of the matrix)

Diagonalization

A matrix A is diagonalizable if there are enough linearly independent eigenvectors to form a basis