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.

Test your understanding of key concepts in Linear Algebra, including matrix rank, linear combinations, linear dependence, and the properties of matrices. This quiz will help solidify your grasp of foundational topics essential for advanced studies in mathematics.