# Linear Algebra

Problem 1

Find an example of a square matrix $$A$$ with real entries such that $$A^2=0$$ but $$AA^T\neq 0$$.

Problem 2

Assume that $$A$$ is an $$n\times n$$ matrix with real entries $$a_{ij}$$ for $$i,j\in\{1,2,\dots n\}$$ such that $$a_{ij}=0$$ if $$i\geq j$$. Prove that $$A^n=0$$.

Problem 3

Assume that $$A$$ and $$B$$ are two quadratic matrices such that $$AB=BA$$. Prove that if $$x$$ is an eigenvector for $$A$$, then $$Bx$$ is an eigenvector for $$A$$ as well.

Problem 4

Solve the system of recursive equations: \begin{eqnarray*} a_{n+1}&=& 2a_n+2b_n\\ b_{n+1}&=& 3a_n + c_n\\ c_{n+1}&=& -3a_n+c_n, \end{eqnarray*} with the initial conditions $$a_0=2$$, $$b_0=-1$$, $$c_0=1$$.

Problem 5

Let $$A$$ and $$B$$ be linear transformations on a finite dimensional vector space $$V$$. Prove that $$\dim \ker (AB)\leq \dim\ker (A)+\dim\ker (B)$$.

$$\ker (A)$$ denotes the kernel of $$A$$, i.e. $$\ker (A)=\{v: A(v)=0\}$$, and $$\dim V$$ denotes the dimension of the vector space $$V$$.

Problem 6

Given an $$n\times n$$ matrix $$M$$ with complex entries, let us denote by $$M^*$$ its conjugate transpose defined as $$M^*=\overline{M}^T$$. The matrix $$M$$ is called normal if $$AA^*=A^*A$$. A matrix $$U$$ is called unitary if $$U^*=U^{-1}$$.

• (a) Prove that if $$M$$ is any complex matrix, then there exists a unitary matrix $$U$$ and an upper-triangular matrix $$D$$ such that $$M=UDU^*$$.

• (b) Prove that if $$M$$ is a normal matrix, then $$D$$ from part (a) is diagonal.

Problem 7

Does there exist an $$n\times n$$ real matrix $$A$$ such that $$\mbox{tr }(A)=0$$ and $$A^2+A^T=I$$? ($$\mbox{tr }(A)$$ is the trace of the matrix $$A$$, and $$A^T$$ is its transpose).

Problem 8

Let $$\delta > 0$$ be given. Assume that $$(A_n)_{n=1}^{\infty}$$ is a sequence of subsets of a unit cube such that the volume of each $$A_n$$ is greater than or equal to $$\delta$$. Prove that for each $$\varepsilon > 0$$ there are integers $$i$$ and $$j$$ such that the volume of $$A_i\cap A_j$$ is at least $$\delta^2-\varepsilon$$.