MATH 4242 Midterm 1 Study Guide

Midterm 1 will cover sections 1.1-1.6, 1.8, 1.9, and 2.1-2.5 of Olver and Shakiban.

Conceptual Skills • Reason using the equivalence between the following statements about an n × n matrix A: – A is non-singular – A can be reduced to upper triangular form with nonzero diagonal elements using elementary row operations 1 (adding a multiple of one row to another) and 2 (swapping rows) (1.4) – A has n nonzero pivots (1.4) – A−1 exists (1.5) – Ax = b has a unique solution for every choice of b (1.5) – A has rank n (1.8) – det A 6= 0 (1.9) – ker(A) = 0 (2.5)

– img(A) = Rn (2.5) • Given information about one of the following properties of an linear system of m equations in n unknowns, Ax = b, deduce information about another: – the row echelon form of [A|b] – the dimension of the kernel or image of A (2.5) – the number of solutions to the linear system Ax = b or Ax = 0 • Describe the beneﬁts of Gaussian Elimination followed by Back Substitution for computing solutions to systems of linear equations instead of computing matrix inverses (1.5)

m • Given a set of vectors {v1 ··· vn} ⊂ R , reason about relationships between their span, linear (in)dependence, and whether they form a basis for a given space. (2.3, 2.4)

m • Given a set of vectors {v1 ··· vn} ⊂ R , relate their span, linear (in)dependence, and basis to properties of the m × n matrix v1 ··· vn (2.3, 2.4, 2.5) • Use the Superposition Principle to ﬁnd a solution to a linear system (2.5) • Use the Fundamental Theorem of Linear Algebra to reason about the dimensions of the kernel, image, cokernel, and coimage of a matrix (2.5)

Procedural Skills • Perform Gaussian Elimination to reduce a given m × n matrix to row echelon form (1.3, 1.4) • Use Gaussian Elimination and Back Substitution to solve a linear system of equations (1.3, 1.4) • Write an elementary matrix E such that performing a desired elementary row operation on A is equivalent to multiplying EA. (1.3, 1.4, 1.5) • Compute the LU factorization of a 2 × 2 or 3 × 3 matrix. (1.3) • Determine whether a set of vectors is linearly independent (2.3) • Determine whether a vector is in the span of a set of vectors (2.3) • Determine whether a set of vectors spans a given vector space (2.3) • Calculate the kernel, image, cokernel, and coimage of a matrix (2.5) • When the matrix A is singular, compute the general solution to the system Ax = b (2.5)

1 Deﬁnitions and Facts about Matrices and Vector Spaces • Deﬁne...

– matrix descriptors: zero, identity, diagonal, upper triangular, lower triangular, symmetric, permutation, elementary – the inverse A−1 of a square matrix A (1.5) – the transpose AT of a matrix A (1.6) – the rank of a matrix A (1.8) – span of a set of vectors (2.3) – linear (in)dependence (2.3) – a basis for a vector space (2.4) – the standard basis (2.4) – the dimension of a vector space (2.4) – the kernel, range, co-kernel, and co-range of a matrix A (2.5) • Compare the rules of matrix arithmetic and real number arithmetic (1.2) • Identify true and false statements about matrix products, inverses, transposes, and determinants; if false, provide a counterexample. (1.2, 1.5, 1.6, 1.9) • Determine whether a given set, with given addition and scalar multiplication rules, forms a vector space (2.1) • Determine whether a given subset of a vector space forms a vector subspace (2.2)

• Find bases of common spaces, e.g. P(n), or the set of solutions to a homogeneous system of diﬀerential equations. (2.4)