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• Find a Basis For the Null Space of a Given $2\times 3$ Matrix Let \[A=\begin{bmatrix} 1 & 1 & 0 \\ 1 &1 &0 \end{bmatrix}\] be a matrix. Find a basis of the null space of the matrix $A$. (Remark: a null space is also called a kernel.)   Solution. The null space $\calN(A)$ of the matrix $A$ is by […]
• Exponential Functions are Linearly Independent Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers. Show that exponential functions \[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}\] are linearly independent over $\R$. Hint. Consider a linear combination \[a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.\] […]
• The Polynomial Rings $\Z[x]$ and $\Q[x]$ are Not Isomorphic Prove that the rings $\Z[x]$ and $\Q[x]$ are not isomoprhic.   Proof. We give three proofs. The first two proofs use only the properties of ring homomorphism. The third proof resort to the units of rings. If you are familiar with units of $\Z[x]$, then the […]
• Simple Commutative Relation on Matrices Let $A$ and $B$ are $n \times n$ matrices with real entries. Assume that $A+B$ is invertible. Then show that \[A(A+B)^{-1}B=B(A+B)^{-1}A.\] (University of California, Berkeley Qualifying Exam) Proof. Let $P=A+B$. Then $B=P-A$. Using these, we express the given […]
• Every Prime Ideal in a PID is Maximal / A Quotient of a PID by a Prime Ideal is a PID (a) Prove that every prime ideal of a Principal Ideal Domain (PID) is a maximal ideal. (b) Prove that a quotient ring of a PID by a prime ideal is a PID.   Proof. (a) Prove that every PID is a maximal ideal. Let $R$ be a Principal Ideal Domain (PID) and let $P$ […]
• Find the Limit of a Matrix Let \[A=\begin{bmatrix} \frac{1}{7} & \frac{3}{7} & \frac{3}{7} \\ \frac{3}{7} &\frac{1}{7} &\frac{3}{7} \\ \frac{3}{7} & \frac{3}{7} & \frac{1}{7} \end{bmatrix}\] be $3 \times 3$ matrix. Find \[\lim_{n \to \infty} A^n.\] (Nagoya University Linear […]
• $x^3-\sqrt{2}$ is Irreducible Over the Field $\Q(\sqrt{2})$ Show that the polynomial $x^3-\sqrt{2}$ is irreducible over the field $\Q(\sqrt{2})$.   Hint. Consider the field extensions $\Q(\sqrt{2})$ and $\Q(\sqrt[6]{2})$. Proof. Let $\sqrt[6]{2}$ denote the positive real $6$-th root of of $2$. Then since $x^6-2$ is […]
• An Example of a Matrix that Cannot Be a Commutator Let $I$ be the $2\times 2$ identity matrix. Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$.   Proof. Assume that $[A, B]=-I$. Then $ABA^{-1}B^{-1}=-I$ implies \[ABA^{-1}=-B. […]