Linear Algebra Concepts Quiz

Rank of a Matrix

• Rank represents the number of non-zero rows in any row-echelon form of matrix A.
• Denoted as rank(A).

Linear Combination of Vectors

• A vector v can be expressed as a linear combination of vectors v1, v2,..., vR using scalars c1, c2,..., cR such that v = c1v1 + c2v2 +...+ cRvR.

Linear Dependence

• A set of vectors v1, v2,..., vR in Rn is linearly dependent if there exist scalars c1, c2,..., cR, not all zero, such that c1v1 + c2v2 +...+ cRvR = 0.
• If no such scalars exist, the set is independent.

Inverse of a Matrix

• An nxn matrix A has an inverse A' if A'A = AA' = I, where I is the identity matrix.

Elementary Matrices

• Elementary matrices are formed from the identity matrix by performing a single row operation.

Subspace

• A subspace of Rn must satisfy:
  • Includes the zero vector (0 ∈ S).
  • Closed under addition (if u, v ∈ S, then u + v ∈ S).
  • Closed under scalar multiplication (if u ∈ S and c ∈ R, then cu ∈ S).

Basis

• A basis for a subspace S of Rn consists of vectors that both span S and are linearly independent.

Dimension of a Subspace

• The dimension of a subspace S is determined by the number of vectors in a basis for that subspace.

Linear Transformation

• A mapping T: Rn → Rm is a linear transformation if it satisfies T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for any scalars c1, c2 and vectors v1, v2.

Eigenvalues and Eigenvectors

• For an nxn matrix A, a scalar λ is an eigenvalue if there exists a non-zero vector x such that Ax = λx.
• Such a vector x is called an eigenvector corresponding to λ.

Similar Matrices

• Matrices A and B are similar (A ~ B) if there exists an invertible matrix P such that P⁻¹AP = B.

Diagonalizable Matrices

• An nxn matrix A is diagonalizable if there is a diagonal matrix D such that A is similar to D via an invertible nxn matrix P (P⁻¹AP = D).

Null Space

• The null space of an mxn matrix A consists of all solutions to the homogeneous equation Ax = 0, denoted as null(A).

Row Space

• The row space of an mxn matrix A is the subspace of Rn spanned by the rows of A.

Column Space

• The column space of an mxn matrix A is the subspace of Rm spanned by the columns of A.

Nullity of a Matrix

• Nullity is the dimension of the null space of matrix A, denoted as nullity(A).

Standard Matrix of a Linear Transformation

• For a linear transformation T: Rn → Rm, the standard matrix A can be represented as A = [T(e1), T(e2), ..., T(en)], where e1, e2, ..., en denote the standard basis vectors.

Characteristic Polynomial

• The characteristic polynomial is found by expanding det(A - λI) to determine eigenvalues.

Characteristic Equation

• The characteristic equation is expressed as det(A - λI) = 0.

Eigenspace

• The eigenspace corresponding to an eigenvalue λ contains all eigenvectors associated with λ, including the zero vector, denoted as E(λ).

Algebraic Multiplicity

• The algebraic multiplicity of an eigenvalue refers to its occurrence as a root in the characteristic equation.

Geometric Multiplicity

• The geometric multiplicity is defined as the dimension of the eigenspace corresponding to an eigenvalue.

The Basis Theorem

• Any two bases for a subspace S have the same number of vectors.

The Rank Theorem

• For an mxn matrix A, the relationship rank(A) + nullity(A) = n holds true.

The Fundamental Theorem of Invertible Matrices

• An nxn matrix A is invertible if and only if:
  • Ax = b has a unique solution for every b in Rn.
  • Ax = 0 has only the trivial solution.
  • The reduced row echelon form of A equals the identity matrix I.
  • A can be expressed as a product of elementary matrices.
  • rank(A) = n.
  • nullity(A) = 0.