a set of preliminary LA exercises for game dev

Game/Levels/Tmp/LinearAlgebra/prelim_exercises.md

@@ -0,0 +1,75 @@
+## Finite Sets
+
+Perhaps a good boss-level exercise would be the theorem 2.1.4 from Fischer's Linear Algebra.
+
+Theorem:
+Sind $X$ und $Y$ endliche Mengen mit gleich vielen Elementen, so sind für eine  Abbildung $f : X \to Y$ folgende Bedingungen äquivalent:
+i) f ist injektiv,
+ii) f ist surjektiv,
+iii) f ist bijektiv.
+
+Remark: This theorem admits different proofs (e.g. proof by contradiction using the pigeonhole principle, proof by induction, etc). These different proofs use different APIs of finite sets.
+
+## Union of Subspaces
+Prove that the union of two subspaces of a vector space is a subspace if and only if one of the
+subspaces is contained in the other.
+
+Remark: This may not be suitable for a boss level exercise but it is a good exercise to understand the definition of a subspace.
+
+## Infinite Dimensional Vector Space
+
+1. Show that $ℚ^n$ is a finite dimensional vector space over $ℚ$.
+2. Show that if $ℝ$ with its standard addition is a vector space over $ℚ$ then the scalar multiplication is given by the standard multiplication of real numbers.
+3. Show that $ℝ$ with its standard addition is an infinite dimensional vector space over $ℚ$.
+
+Remark: Once of proof of the last part requires uncountability of $ℝ$. Another proof uses the fact that $\log p_i$ are linearly independent over $ℚ$ where $p_i$ are prime numbers.
+
+## Proper Subspaces of $n \times n$ Matrices
+1. Show that the space of $n \times n$ matrices with real entries is a vector space over $ℝ$.
+2. Suppose $A$ is an $n \times n$ matrix with real entries. Show that the space generated by the powers of $A$, i.e. the set $\{I, A, A^2, A^3, \ldots\}$ is a proper subspace of the space of $n \times n$ matrices with real entries.
+3. Show that a vector space over an infinite field cannot be a finite union of proper subspaces.
+
+## The lattice of subspaces of a vector space and the modular law
+1. Show that the subspaces of a vector space form a lattice under the inclusion order.
+2. Show that the lattice of subspaces of a vector space satisfies the modular law.
+3.
+
+
+## Van der Monde Matrix
+Fix $100$ distinct points $t_0, \ldots, t_{99}$ in the interval  $I = [ -1 ,1 ]$. Consider the map $L \colon \mathbb{R}^{200} \to \mathbb{R}^{100}$ defined by the assignment
+$$ c = (c_0,...,c_{199}) \mapsto (p_c(t_0),...,p_c(t_{99})) $$
+where  $p_c = \sum_{i=0}^{i=199} c_i x^i$, i.e. from vectors of coefficients of polynomials of degree $\leq 199$ to the vectors $(p_c(t_i))_{i=0}^{99}$ of values of such polynomials at nodes $t_i$.
+
+
+1. Show that $L$ is linear.
+2. Show that this map is represented, upon choosing the standard basis in $\mathbb{R}^{200}$ and $\mathbb{R}^{100}$, by the $100 \times 200$ _Vendermonde_ matrix.
+
+$$ \begin{bmatrix}
+1 & t_0 & t_0^2 & \dots & t_0^{199}\\
+1 & t_1 & t_1^2 & \dots & t_1^{199}\\
+1 & t_2 & t_2^2 & \dots & t_2^{199}\\
+\vdots & \vdots & \vdots & \ddots &\vdots \\
+1 & t_{99} & x_{99}^2 & \dots & t_{99}^{199}
+\end{bmatrix} $$
+
+3. Show that this map is never invertible.
+
+
+## Quotients
+1. Construction of the field $𝔽_p$ via quotient construction of its underlying cyclic group for a prime number p.
+2. (Advanced) (Gaussian Coefficients) Let $𝔽$ be a finite field of size $q$ and let $V$ be an $n$-dimensional vector space over $𝔽$. Show that the number of subspaces of $V$ is given by the Gaussian Coefficients.
+
+TODO: I will break down the question 2 into four separate parts (Sina.)
+
+## Trace
+1. Show that the trace of a matrix, the sum of its diagonal entries, is a linear map from the space $n \times n$ matrices to the field of scalars.
+2. Show that $\mathrm{tr} (A A^T) \ge 0$ and the equality holds if and only if $A = 0$.
+3. Suppose $f$ is a linear transformation over the space of $n \times n$ matrices such that $f(A B) = f(B A)$ for all $A, B$. Show that there exists a scalar $c$ such that $f(A) = c \mathrm{tr}(A)$ for all $A$.
+4. Show that for any matrix $A$ the map $\mathrm{tr}(A \; \bullet) : X \mapsto \mathrm{tr}(AX)$ is a linear map on the space of $n \times n$ matrices.
+5. Show that the map above is a zero map if and only if $A = 0$.
+6. Show that the map $A \mapsto \mathrm{tr}(A \; \bullet)$ is an isomorphism.
+7. Use (4) to prove that every linear map on the space of $n \times n$ matrices is of the form $\mathrm{tr}(A \; \bullet)$ for some matrix $A$.
+8. Suppose $A$ is an $m \times n$ and $B$ is an $n \times m$ matrix. Show that the trace of $AB$ is the same as the trace of $BA$.
+9. Show that the trace of a matrix is the sum of its eigenvalues.
+
+## Singular Value Decomposition (?)