Numerical Linear Algebra (Spring 2021) Homework: 01
ICSI 501 – Numerical Linear Algebra
Instructor: Abram Magner Date: Spring 2021
Instructions: Please answer the following questions in complete sentences, showing all work
(including code and output of programs, when applicable). I prefer that you type your solutions
(e.g., using LaTeX, with Overleaf, TeXworks, etc., or Word), but will accept handwritten notes. If
the grader cannot read your handwriting, then they cannot award you points.
Due date: Monday, 3/1/2021, 11:59 p.m., on Blackboard.
The problems below use concepts learned in class and in the course prerequisites. They are not
meant to be super-tricky, but they do require an understanding of these concepts.
1.1 Fields
1. Let F be a field. Prove that F cannot have two different additive identities (i.e., there cannot
be two different elements a, b ∈ F such that, for every x ∈ F, a+x = x+a = x = b+x = x = b).
Hint: Prove this by contradiction.
1.2 Bases
1. Let Rm×n denote the space of m × n matrices with real entries. This is a vector space over
R, with the usual operations.
Exhibit a basis for this space.
2. Determine the dimension of Rm×n.
1.3 Elementary row operations
1. Consider the following function on Rm R: for a vector v ∈ Rm, f(v) swaps elements i and j.
Show that f : Rm → Rm is a linear function.
2. Exhibit a matrix for f with respect to the standard basis for Rm.
1.4 Absolute and relative error
1. Consider a sequence an of real numbers such that limn→∞ an = 0. Suppose that we approxi-
mate an with aˆn. Is it possible for the absolute error in approximating an with aˆn to converge
1-1
1-2 Homework 01: ICSI 501 – Numerical Linear Algebra
to 0 while the relative error tends to ∞? If so, exhibit a sequence an and a sequence aˆn for
which this happens. If not, prove it.
1.5 Rank and nullity
1. Recall that the rank of an m×n matrix M is defined to be the dimension of the space spanned
by its columns (so it’s the maximum number of linearly independent columns of M). Suppose
M is m ×m (i.e., a square matrix) and that its columns form a basis for Rm. What is the
rank of M? Prove your answer.
2. Suppose that A ∈ Rm×n and x ∈ Rn is nonzero, satisfying Ax = 0 (that is, x is in the kernel
of A). Are the columns of A linearly independent? Prove your answer.

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