UMBC MATH 221 Exam 2 2026 (200+ Questions) – Linear Algebra – Invertible Matrix Theorem, LU Factorization, Subspaces, Rank & Dimension Q&A

This document contains over 200 fully answered exam questions for UMBC MATH 221 Exam 2 2026, focusing on advanced Linear Algebra concepts including the Invertible Matrix Theorem, LU factorization, subspaces, column space, null space, rank, dimension, basis, and vector space theory. The material outlines the equivalent statements for an invertible n×n matrix, including row equivalence to the identity matrix, n pivot positions, linearly independent columns, Ax = 0 having only the trivial solution, and the linear transformation x ↦ Ax being one-to-one and onto. The study guide provides detailed explanations of invertibility conditions for square matrices, including implications such as AB = I meaning both A and B are invertible and are inverses of each other. It connects invertible matrices to linear transformations, explaining that T is invertible if and only if its standard matrix A is invertible, and that the inverse transformation is given by S(x) = A⁻¹x. LU factorization is covered in depth, with step-by-step procedures for solving Ax = b by decomposing A into a lower triangular matrix L and an upper triangular matrix U, then applying forward and backward substitution. Extensive sections address subspaces of Rⁿ, including the three defining properties (contains the zero vector, closed under addition, closed under scalar multiplication). The document defines and distinguishes the column space (Col A), null space (Nul A or Ker A), and row space, and emphasizes that the pivot columns form a basis for the column space. It introduces rank as the dimension of the column space and presents the Rank Theorem: dim(Nul A) + rank(A) = n. Vector space theory is thoroughly examined, including definitions of basis, coordinates relative to a basis, dimension, spanning sets, linear independence, and isomorphisms. The material highlights key theorems stating that any linearly independent set of p elements in a p-dimensional space is automatically a basis, that every basis of a finite-dimensional vector space contains the same number of vectors, and that any linearly independent set in a subspace can be extended to a basis. Relationships between dimension, pivot columns, and free variables are clearly explained. The study guide concludes with proofs involving linear transformations, demonstrating how linear dependence is preserved under transformations and establishing that T is one-to-one if and only if T(v₁) = T(v₂) implies v₁ = v₂, equivalently ker(T) = {0}. This document is particularly relevant for: UMBC MATH 221 students preparing for Exam 2 Undergraduate students enrolled in Linear Algebra STEM majors studying vector spaces and matrix theory Engineering and Computer Science students reviewing rank and dimension Students preparing for midterm or unit assessments in linear algebra It is suitable for courses such as: Linear Algebra I Matrix Theory and Applications Vector Spaces and Linear Transformations Applied Linear Algebra for Engineers Mathematical Foundations for Data Science Keywords: UMBC MATH 221 exam 2 2026, invertible matrix theorem equivalent statements, LU factorization forward backward substitution, column space null space row space, rank plus nullity theorem, basis and dimension vector space, pivot columns basis col A, one to one linear transformation ker T equals zero, onto transformation span Rn, coordinates relative to basis, isomorphism linear algebra, AB equals I invertible matrices, finite dimensional vector space theorems, extend linearly independent set to basis