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.

Test your knowledge on key concepts for your Linear Algebra final exam. This quiz features essential definitions such as consistent systems of equations and linear dependence and independence of vectors. Review and prepare effectively for your upcoming assessment!