---
title: Assignments 5
tags: Linear algebra
GA: G-77TT93X4N1
---

# Assignments 5

1. Let $V$ be an inner product space spanned by the basis $\beta=\{1, \sin (\pi x), \cos (\pi x), e^{-x}\}$ with inner product
$$
\langle f, g\rangle = \int_{-1}^1 f(x)g(x)\,dx.
$$
Let $T:V\to V$ be defined by $T(f) = f'(x)$, the derivative of $f$ with respect to $x$.
* Determine a basis for each of the four fundamental subspaces of $T$.
2. Let $V$ be the vector space of $n\times n$ matrices, $n\in\mathbb{N}$, $n\ge 2$. Given $A\in V$, we define the linear transformation $T\in \mathcal{L}(V)$ by $T(X) = AX - XA$.
* Show that $\text{dim}(\text{Kernel}(T))\ge 2$.
3. Let $V$ be an inner product space spanned by the basis $\beta=\{1, x, x^2\}$ with inner product
$$
\langle f, g\rangle = \int_{-1}^1 x^2f(x)g(x)\,dx.
$$
Let $T:V\to V$ be defined by $T(f) = xf'(x)$.
* Determine the matrix representation of $T$, denoted as $[T]_{\beta}$.
* Let us define a linear transformation $\tilde{T}:V\to V$ with the matrix representation $[\tilde{T}]_\beta = \text{transpose}([T]_{\beta})$. Find $f, g\in V$ such that
$$
\langle T(f), g\rangle \ne \langle f, \tilde{T}(g)\rangle.
$$
This result indicate that $\tilde{T}$ is not the adjoint transformation of $T$.
4. Let $V$ be an inner product space spanned by the basis $\beta=\{1, x, x^2\}$ with inner product
$$
\langle f, g\rangle = \int_{-1}^1 x^2f(x)g(x)\,dx.
$$
Let $T:V\to V$ be defined by $T(f) = xf'(x)$.
* Determine $T^*:V\to V$.
5. Let $V$ be an inner product space spanned by the basis $\beta=\{1, x, x^2\}$ with inner product
$$
\langle f, g\rangle = \int_{-1}^1 x^2f(x)g(x)\,dx.
$$
Let $T:V\to V$ be defined by $T(f) = xf'(x)$.
* Find all the eigenvalues and eigenvectors of $T$.

# TA solution

* [Assignment5_solution](https://drive.google.com/file/d/1ZmLDUxmw2Ohrj7RFjt6KyvbYgeH0HkNO/view?usp=sharing)