# number2018

by Yu ·

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- The Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt Orthogonalization Consider the $2\times 2$ real matrix \[A=\begin{bmatrix} 1 & 1\\ 1& 3 \end{bmatrix}.\] (a) Prove that the matrix $A$ is positive definite. (b) Since $A$ is positive definite by part (a), the formula \[\langle \mathbf{x}, […]
- The Range and Nullspace of the Linear Transformation $T (f) (x) = x f(x)$ For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis. Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, […]
- Example of an Element in the Product of Ideals that Cannot be Written as the Product of Two Elements Let $I=(x, 2)$ and $J=(x, 3)$ be ideal in the ring $\Z[x]$. (a) Prove that $IJ=(x, 6)$. (b) Prove that the element $x\in IJ$ cannot be written as $x=f(x)g(x)$, where $f(x)\in I$ and $g(x)\in J$. Hint. If $I=(a_1,\dots, a_m)$ and $J=(b_1, \dots, b_n)$ are […]
- Find All the Eigenvalues of 4 by 4 Matrix Find all the eigenvalues of the matrix \[A=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 &0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}.\] (The Ohio State University, Linear Algebra Final Exam Problem) Solution. We compute the […]
- Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$ Consider the complex matrix \[A=\begin{bmatrix} \sqrt{2}\cos x & i \sin x & 0 \\ i \sin x &0 &-i \sin x \\ 0 & -i \sin x & -\sqrt{2} \cos x \end{bmatrix},\] where $x$ is a real number between $0$ and $2\pi$. Determine for which values of $x$ the […]
- An Example of a Real Matrix that Does Not Have Real Eigenvalues Let \[A=\begin{bmatrix} a & b\\ -b& a \end{bmatrix}\] be a $2\times 2$ matrix, where $a, b$ are real numbers. Suppose that $b\neq 0$. Prove that the matrix $A$ does not have real eigenvalues. Proof. Let $\lambda$ be an arbitrary eigenvalue of […]
- Mathematics About the Number 2017 Happy New Year 2017!! Here is the list of mathematical facts about the number 2017 that you can brag about to your friends or family as a math geek. 2017 is a prime number Of course, I start with the fact that the number 2017 is a prime number. The previous prime year was […]
- Linearly Independent/Dependent Vectors Question Let $V$ be an $n$-dimensional vector space over a field $K$. Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ are linearly independent vectors in $V$. Are the following vectors linearly independent? \[\mathbf{v}_1+\mathbf{v}_2, \quad \mathbf{v}_2+\mathbf{v}_3, […]