Linear Algebra Concepts

Flashcards on key linear algebra concepts including invertibility, diagonalizability, and eigenvectors.

Invertibility

A matrix is invertible if and only if $0$ is NOT an eigenvalue; it has full rank, a non-zero determinant, and zero free variables.

Diagonalizability

A matrix is diagonalizable if you can find enough independent eigenvectors to form a basis; if an eigenvalue has an algebraic multiplicity of 2 but only 1 basis vector, it is NOT diagonalizable.

Eigenspace

The eigenspace of a matrix is the null space of the shifted matrix $(A - au I)$ for a given eigenvalue $ au$.

Eigenspace Dimension

The dimension of an eigenspace is determined by the number of free variables, not pivots.

Row Operations and Eigenvalues

Row operations destroy eigenvalues; they preserve null space and row space but change eigenvalues and determinant.

Free Variables

When finding eigenvectors, distinct free variables must be separated into different basis vectors.

Diagonalization Matrices Order

The columns of the eigenvector matrix $P$ must match the order of the eigenvalues in matrix $D$ when building diagonalization.

Right-to-Left Composition

For the standard matrix of $S ackslash circ T$, the operation $T$ is applied first, thus matrices are multiplied right-to-left: $S imes T$.

Cofactor Expansion

When performing cofactor expansion for determinants, always include brackets: $ ext{Scalar} imes [(ad)-(bc)]$.

Testing Eigenvectors

To identify an eigenvector, multiply the matrix by given vectors; if the result is a scalar multiple of the original vector, it is an eigenvector.

Stop at REF

To find bases for the Column, Row, and Null spaces, row-reduce to Row Echelon Form (REF) only; RREF is not required.

The Zero Hunter

For $4 imes 4$ determinants, perform row additions to create a column with three zeros for easier expansion.