Homework 0 — Vector Spaces

Learning objectives

Verify a structure is a vector space by checking essential axioms efficiently.
Exhibit and justify infinite linear independence within a function space.
Describe distinct bases for the same space and compute dimensions.
Apply reasoning to prove linear independence of function families.

Rubric: Correctness, Clarity/structure, Precision of definitions.

Warm-up self-check

Q1. Which set is not a vector space over \(\mathbb{R}\) under pointwise operations?

\(C[0,1]\) (continuous functions)
\(C^1[0,1]\) (continuously differentiable)

Strictly positive continuous functions \(\{f>0\}\)
Polynomials

Problem 0 Real-world vector space

Give a real-world example of a collection of objects that forms a vector space over \(\mathbb{R}\) under natural operations.

Define the set \(S\) precisely (what are the objects?).
Specify the operations: addition \(+\) and scalar multiplication \(\alpha\cdot\), and state the field.
Verify two vector-space axioms in full detail (your choice), and briefly state why the remaining axioms hold.
Explain in 2–3 sentences why these operations are “natural” in the context you chose.

Problem 1 Vector space verification

Prove that \(C[0,1]\) with pointwise addition and scalar multiplication is a vector space over \(\mathbb{R}\).

Proof plan (check as you go):

State the field and the operations (pointwise).
Identify the zero vector \(0(t)\equiv 0\) and additive inverse.
Show closure under \(+\) and scalar multiplication.
Justify two representative axioms (associativity; distributivity); the rest follow pointwise from \(\mathbb{R}\).

Hint (nudge)

Treat \(C[0,1]\subset \mathbb{R}^{[0,1]}\) (this notation stands for the vector space of real-valued functions defined on the compact set \([0,1]\); pointwise operations are inherited from \(\mathbb{R}\), a field.

Toy counterexample (what fails and why)

Strictly positive continuous functions are not a vector space: not closed under negative scalar multiples.
Polynomials with leading coefficient 1 are not closed under addition.

Problem 2 Infinite dimension

Prove that \(C[0,1]\) is infinite-dimensional.

Consider the following set of monomials \(\{1,x,x^2,\dots,x^n\}\) (prove by induction using the \( \epsilon\)-\(\delta\) definition that every element in this set is continuous on the closed interval \([0,1]\)). Next, suppose a dependence \(\sum_{k=0}^n a_k x^k=0\) and proceed from there.

Hint

A nontrivial polynomial of degree \(n)\ has at most \(n\) roots; choose \(n+1\) points (why?)

Problem 3 Bases & dimension

Let
\( V = M_2(\mathbb{C}) := \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \;:\; a,b,c,d \in \mathbb{C} \right\} \),
equipped with entrywise addition and scalar multiplication.
Find 3 distinct bases for \(V\) and compute \(\dim_{\mathbb{C}} V\).

Standard units \(E_{11},E_{12},E_{21},E_{22}\) form a basis; \(\dim V=4\). Extension: give an alternative basis (identity + three traceless matrices) to illustrate non-uniqueness.

Problem 4 Linear independence of Gaussian translates

Let \(g(t)=e^{-t^2}\), \((T_x g)(t)=g(t-x)\), and \(\Lambda\subset\mathbb{R}\) finite with distinct points. Show \(\{T_\lambda g: \lambda\in\Lambda\}\) is linearly independent in \(C(\mathbb{R})\).

What to submit

Handwritten solutions (paper or tablet with a stylus), neat and legible or typed up in latex (if you want to practice acquiring this skill)
For each problem: clearly state any notation/definitions you use; give a short plan (1–2 lines); then a complete, logically structured proof.
Justify nontrivial steps and name theorems you invoke. Avoid phrases like “obvious” without explanation.
If you consulted any resources or peers, list them briefly. The write‑up must be your own words.

Rubric: Correctness, Structure & clarity, Precision of notation.