Purdue-univeristy-exam-eye-catch
Purdue-univeristy-exam-eye-catch
Add to solve later
Sponsored Links
Add to solve later
Sponsored Links
More from my site
- Pullback Group of Two Group Homomorphisms into a Group
Let $G_1, G_1$, and $H$ be groups. Let $f_1: G_1 \to H$ and $f_2: G_2 \to H$ be group homomorphisms.
Define the subset $M$ of $G_1 \times G_2$ to be
\[M=\{(a_1, a_2) \in G_1\times G_2 \mid f_1(a_1)=f_2(a_2)\}.\]
Prove that $M$ is a subgroup of $G_1 \times G_2$.
[…]
- Is the Derivative Linear Transformation Diagonalizable?
Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by
\[ T( ax^2 + bx + c ) = 2ax + b . \]
Is $T$ diagonalizable? If so, find a diagonal matrix which […]
- Algebraic Number is an Eigenvalue of Matrix with Rational Entries
A complex number $z$ is called algebraic number (respectively, algebraic integer) if $z$ is a root of a monic polynomial with rational (respectively, integer) coefficients.
Prove that $z \in \C$ is an algebraic number (resp. algebraic integer) if and only if $z$ is an eigenvalue of […]
- Prove that the Dot Product is Commutative: $\mathbf{v}\cdot \mathbf{w}= \mathbf{w} \cdot \mathbf{v}$
Let $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors.
(a) Prove that $\mathbf{v}^\trans \mathbf{w} = \mathbf{w}^\trans \mathbf{v}$.
(b) Provide an example to show that $\mathbf{v} \mathbf{w}^\trans$ is not always equal to $\mathbf{w} […]
- A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space
(a) Suppose that $A$ is an $n\times n$ real symmetric positive definite matrix.
Prove that
\[\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}\]
defines an inner product on the vector space $\R^n$.
(b) Let $A$ be an $n\times n$ real matrix. Suppose […]
- A One-Line Proof that there are Infinitely Many Prime Numbers
Prove that there are infinitely many prime numbers in ONE-LINE.
Background
There are several proofs of the fact that there are infinitely many prime numbers.
Proofs by Euclid and Euler are very popular.
In this post, I would like to introduce an elegant one-line […]
- Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$
Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.
(a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.
(b) Let
\[A^{100}=aA^2+bA+cI,\]
where $I$ is the $3\times 3$ identity matrix.
Using the […]
- Elements of Finite Order of an Abelian Group form a Subgroup
Let $G$ be an abelian group and let $H$ be the subset of $G$ consisting of all elements of $G$ of finite order. That is,
\[H=\{ a\in G \mid \text{the order of $a$ is finite}\}.\]
Prove that $H$ is a subgroup of $G$.
Proof.
Note that the identity element $e$ of […]