Linear Algebra Concepts

Null Space and Related Theorems

• Null space (Nul A) is the set of all solutions to the homogeneous equation Ax=0.
• Nul A is a subspace of Rⁿ, defined as Nul(A) = {x | Ax=0}, where x is in Rⁿ.
• Null space theorem is important in understanding the solution space of linear equations.

Subspaces

• A subspace H in Rⁿ must contain the zero vector, support closure under addition, and allow scalar multiplication.
• For each u and v in H, u + v must also be in H; for each u in H and scalar c, cu must be in H.

Column Space

• Column space (Col A) consists of all linear combinations of the columns of matrix A.

Standard Basis

• The set {e₁, ..., eη} represents the standard basis for Rⁿ, providing a fundamental set of vectors.

Pivot Columns

• The pivot columns of a matrix A form a basis for its column space, confirming their significance in linear independence.

Diagonalization Theorem

• An nxn matrix A is diagonalizable if it has n linearly independent eigenvectors.
• This can be represented as A = PDP⁻¹, where D is a diagonal matrix containing eigenvalues corresponding to eigenvectors in P.

Dimension of Subspaces

• The dimension of a nonzero subspace H is determined by the number of vectors in any basis for H.
• The dimension of the zero subspace {0} is defined to be zero.

Rank of a Matrix

• The rank of matrix A is the dimension of its column space, denoted as rank A.
• Rank theorem states that for an n-column matrix A, rank A + dim Nul A = n.

Basis Theorem

• A linearly independent set of exactly p elements in a p-dimensional subspace H is a basis for H.
• Similarly, any set of p elements that spans H is also a basis for H.

Invertible Matrix Theorem

• An n × n matrix A is invertible if its columns form a basis of Rⁿ, satisfying various conditions related to rank and null space.

Eigenvectors and Eigenvalues

• An eigenvector of matrix A satisfies the equation Ax = λx for some scalar λ.
• An eigenvalue corresponds to a nontrivial solution of Ax = λx.

Characteristic Equation

• A scalar λ is an eigenvalue of matrix A if it satisfies the equation det(A - λI) = 0.

Properties of Triangular Matrices

• The eigenvalues of a triangular matrix are the entries on its main diagonal.

Determinants

• A matrix A is invertible if and only if det A ≠ 0.
• Properties include det(AB) = (det A)(det B) and det Aⁿ = det A raised to the power of n.
• Row operations affect determinants: row replacement maintains it, row interchange changes the sign, and row scaling multiplies by the scalar.

Similar Matrices

• Similar n × n matrices have the same characteristic polynomial and eigenvalues, ensuring they share essential properties.

Diagonalizability

• An n × n matrix with n distinct eigenvalues is guaranteed to be diagonalizable.
• The dimension of the eigenspace for eigenvalues plays a crucial role in determining diagonalizability.