TAMU Math 304 Final Flashcards

Matrix Properties and Definitions

• Only invertible matrices can be factored into elementary matrices.
• The associative property of matrices states that (AB)C = A(BC).
• Determinants follow the distributive property: det(AB) = det(A) * det(B).
• A matrix B is similar to matrix A if there exists an invertible matrix P such that B = P⁻¹AP.
• Diagonalizable matrices can be expressed as PDP⁻¹, where D is diagonal and P is invertible.

Determinants and Invertibility

• The characteristic polynomial of a matrix A is given by det(A - λI).
• A matrix A is not invertible if det(A) = 0.

Dimensions and Bases

• The dimension of the null space, dim(N(A)), equals the number of null basis elements.
• The change of basis formula is expressed as B = V⁻¹AV.
• In R³, if {x1, x2, x3} forms a basis, then {x1, x2, x2 + x3} also forms a basis.

Least Squares and Orthogonality

• For the least squares solution, compute AᵀA, then Aᵀb, and solve AᵀAx = Aᵀb for x.
• The Gram-Schmidt Process is used to orthogonalize vectors, starting with u₁ = v₁ and adjusting each subsequent vector by projecting onto previous ones.
• The dot product of vectors u and v is denoted as uᵀv, and u is orthogonal to v if uᵀv = 0.

Orthogonal and Orthonormal Sets

• A set {u₁, u₂, u₃} is orthonormal if it satisfies two conditions: the norm of each vector equals 1, and they are pairwise orthogonal (dot products equal 0).

Singular and Linear Independence

• A singular matrix has no inverse, indicated by a determinant of zero.
• The inverse of the product of two matrices is given by (AB)⁻¹ = B⁻¹A⁻¹.
• To determine linear independence, convert a set of vectors to row echelon form; the number of linearly independent vectors corresponds to the dimension of the space.

Symmetric Matrices

• Symmetric matrices have specific properties:
  • The sum of symmetric matrices is symmetric.
  • The product of symmetric matrices may not be symmetric.
  • Powers of symmetric matrices (A^n) remain symmetric if A is symmetric.
  • The inverse of a symmetric matrix (A⁻¹) is symmetric only if A is symmetric.

Span and Subspaces

• The dimension of span(S) is less than the number of maximum vectors if they are linearly dependent, and at least equal to the minimum number of linearly independent vectors.
• A subset S qualifies as a subspace if it contains the zero vector, is closed under addition, and is closed under scalar multiplication.

Column Space

• To determine if vector V is in the column space of matrix A, attempt to solve the equation Ax = V.