Linear Algebra Final Exam Flashcards

Systems of Equations

• A system is consistent if it has only one solution.

Linear Dependence and Independence

• Vectors are linearly dependent if a linear combination equals zero with at least one non-zero coefficient.
• Vectors are linearly independent if the only way a linear combination equals zero is when all coefficients are zero.

Eigenvectors and Eigenvalues

• An eigenvector x of an nxn matrix A satisfies A(x) = λx for some scalar λ.
• A scalar λ is an eigenvalue of A if there exists a nonzero vector x such that A(x) = λx.

Vector Spaces and Spanning

• A set of vectors spans a vector space V if every vector in V can be expressed as a linear combination of these vectors.

Linear Combinations

• A vector V is a linear combination of vectors V1, V2, ..., Vn if there are scalars x1, x2, ..., xn such that V = x1V1 + x2V2 + ... + xnvn.

Linear Transformations

• A function T: R → R is a linear transformation if it satisfies:
  • T(x+y) = T(x) + T(y) for all x, y ∈ R.
  • T(cx) = cT(x) for all x ∈ R and scalars c ∈ R.

Elementary Row Operations

• The three elementary row operations on matrices are:
  • Replacing one row with the sum of itself and a multiple of another row.
  • Interchanging two rows.
  • Multiplying all entries in a row by a nonzero constant.

Reduced Echelon Form

• A matrix is in reduced echelon form if:
  • All nonzero rows precede any row of zeros.
  • Leading entries appear in a column to the right of leading entries in rows above.
  • Each leading entry is 1.
  • Each leading 1 is the only nonzero entry in its column.

Null Space

• The null space Nul T of a linear transformation T: R → R consists of vectors x ∈ R such that T(x) = 0.

Onto and One-to-One Functions

• A function f: X → Y is onto if for each y ∈ Y, there exists at least one x ∈ X with f(x) = y.
• A function f: X → Y is one-to-one (1-1) if f(r) = f(s) implies r = s for any r, s ∈ X.

Characteristic Polynomial

• The characteristic polynomial p(λ) of an nxn matrix A is defined as p(λ) = det(A - λI).

Vector Norm

• The norm of a vector v = [x1, x2, ..., xn] is defined by ||v|| = √(x1² + x2² + ... + xn²).

Eigenspace

• The eigenspace corresponding to an eigenvalue λ is the set of all solutions to the equation det(A - λI) = 0.

Similar Matrices

• Two nxn matrices A and B are similar if there exists an invertible nxn matrix P such that A = PBP⁻¹.

Diagonalizable Matrices

• An nxn matrix is diagonalizable if it can be expressed as A = PDP⁻¹, where D is a diagonal matrix and P is an invertible matrix.

Orthogonal Vectors

• Two vectors u and v are orthogonal if their dot product equals zero.

Basis for a Vector Space

• A set of vectors B = {V1, V2, ..., Vn} is a basis for a vector space V if B is linearly independent and spans V.