Summary - Very Important

• Homogeneous: $AX=0$

  • Always consistent

  • Trivial solution: $X=0$

  • Non-trivial solution if $\text{rank}(A)<n$

  • Nullity = $n - \text{rank}(A)$ → number of free variables

• Non-homogeneous: $AX=b$

  • Consistent if $\text{rank}(A) = \text{rank}([A|b])$

  • No solution if $\text{rank}(A) \neq \text{rank}([A|b])$

• Rank: Number of linearly independent rows/columns

• Nullity: $n - \text{rank}(A)$

• Dimension of null space → solutions to $AX=0$

• Number of free variables = nullity

System $AX=b$ is consistent iff: \[ \text{rank}(A) = \text{rank}([A|b]) \]

• Unique solution: rank = n (full rank)

• Infinite solutions: rank < n and consistent

• No solution: rank([A|b]) > rank(A)

• Always consistent

• Trivial solution: $X=0$

• Non-trivial solution exists if $\text{rank}(A)<n$

• All solutions are linear combinations of basis vectors of null space

• $\det(A)\neq 0 \Rightarrow$ Full rank, invertible

• $\det(A)=0 \Rightarrow$ Singular, rank $<$ n

• Non-homogeneous system unique solution only if $\det(A)\neq0$

\[ A = LU, \quad L=\text{lower triangular}, \quad U=\text{upper triangular} \]

• Solve $AX=b$ via:

  1. $LY=b$ (forward substitution)

  2. $UX=Y$ (backward substitution)

• $A$ invertible $\Rightarrow$ L and U invertible

• Singular $A \Rightarrow$ at least one zero on U diagonal

• $Av = \lambda v$

• Trace = sum of eigenvalues

• Determinant = product of eigenvalues

• Rank = number of non-zero eigenvalues

• Symmetric matrices → real eigenvalues, orthogonal eigenvectors

• $m<n$: fewer equations → infinite solutions (if consistent)

• $m>n$: more equations → may be inconsistent

• $m=n$: full rank → unique solution possible

• Free variables = $n - \text{rank}(A)$

• Infinite solutions = linear combination of null space basis

• Homogeneous: all solutions = linear combinations of null space vectors

• Compute determinant/rank symbolically

• Solve for parameters:

  • Infinitely many solutions → rank drops

  • Unique solution → full rank

  • No solution → augmented rank $>$ rank

• Columns $a_1,\dots,a_n$ linearly dependent if $\sum c_i a_i = 0, c_i \neq 0$

• Implication: solution may be infinite or none depending on augmented rank

• Sum of singular matrices → may be non-singular

• Sum of non-singular matrices → may be singular

• Zero determinant → each row/column = linear combination of others

• Eigenvalues of symmetric → real

• LU decomposition: forward/backward substitution

Homogeneous, Non-homogeneous; Trivial/non-trivial solutions

Rank = independent rows/columns; Nullity = n - rank

System consistent if rank(A) = rank([A|b])

Always consistent; non-trivial if rank < n

det(A) ≠ 0 → invertible; det(A) = 0 → singular

Solve AX=b via LY=b, UX=Y; L,U invertible if A invertible

$Av = \lambda v$; Symmetric → real eigenvalues, orthogonal eigenvectors

m$<$n → infinite solutions; m>n → may be inconsistent

Free variables = n - rank(A); infinite solutions = linear combinations

Use determinant/rank symbolically to find parameter values for solution type

Columns dependent → solution may be infinite or none depending on augmented rank