# linear-transformation-eye-catch

by Yu · Published · Updated

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- Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$ Let $A$ be an $n\times n$ nonsingular matrix with integer entries. Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$. Hint. If $B$ is a square matrix whose entries are integers, then the […]
- Is the Map $T(f)(x) = f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) \cdot x^3$ a Linear Transformation? Let $C ([0, 3] )$ be the vector space of real functions on the interval $[0, 3]$. Let $\mathrm{P}_3$ denote the set of real polynomials of degree $3$ or less. Define the map $T : C ([0, 3] ) \rightarrow \mathrm{P}_3 $ by \[T(f)(x) = f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) […]
- If the Order is an Even Perfect Number, then a Group is not Simple (a) Show that if a group $G$ has the following order, then it is not simple. $28$ $496$ $8128$ (b) Show that if the order of a group $G$ is equal to an even perfect number then the group is not simple. Hint. Use Sylow's theorem. (See the post Sylow’s Theorem […]
- A Line is a Subspace if and only if its $y$-Intercept is Zero Let $\R^2$ be the $x$-$y$-plane. Then $\R^2$ is a vector space. A line $\ell \subset \mathbb{R}^2$ with slope $m$ and $y$-intercept $b$ is defined by \[ \ell = \{ (x, y) \in \mathbb{R}^2 \mid y = mx + b \} .\] Prove that $\ell$ is a subspace of $\mathbb{R}^2$ if and only if $b = […]
- Difference Between Ring Homomorphisms and Module Homomorphisms Let $R$ be a ring with $1$ and consider $R$ as a module over itself. (a) Determine whether every module homomorphism $\phi:R\to R$ is a ring homomorphism. (b) Determine whether every ring homomorphism $\phi: R\to R$ is a module homomorphism. (c) If $\phi:R\to R$ is both a […]
- If the Kernel of a Matrix $A$ is Trivial, then $A^T A$ is Invertible Let $A$ be an $m \times n$ real matrix. Then the kernel of $A$ is defined as $\ker(A)=\{ x\in \R^n \mid Ax=0 \}$. The kernel is also called the null space of $A$. Suppose that $A$ is an $m \times n$ real matrix such that $\ker(A)=0$. Prove that $A^{\trans}A$ is […]
- How to Use the Cayley-Hamilton Theorem to Find the Inverse Matrix Find the inverse matrix of the $3\times 3$ matrix \[A=\begin{bmatrix} 7 & 2 & -2 \\ -6 &-1 &2 \\ 6 & 2 & -1 \end{bmatrix}\] using the Cayley-Hamilton theorem. Solution. To apply the Cayley-Hamilton theorem, we first determine the characteristic […]
- Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$ Let \[A=\begin{bmatrix} 1 & -14 & 4 \\ -1 &6 &-2 \\ -2 & 24 & -7 \end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix} 4 \\ -1 \\ -7 \end{bmatrix}.\] Find $A^{10}\mathbf{v}$. You may use the following information without proving […]