mathdoge

### Questions (21)

 6 Show that the only endomorphism $\phi$ of $\mathbb{Z}_7 \times \mathbb{Z}_7$ satisfying $\phi^5 = \text{id}$ is the identity. 4 Show the matrix commutes with companion matrix is a polynomial 3 Find sequence limit $x_{n+1} = \frac{3}{4} x_n + \frac{1}{4} \int_0^{|x_n|} f(x) dx$ 2 $f, g: \mathbb{R} \to \mathbb{R}$ and $f(x+h) = f(x) + g(x)h + a(x,h)$ for $|a(x,h)| \leq Ch^3$. Show that $f$ is affine. 2 If the multiplicity of $\lambda_1$ is $r_1$ in characteristic polynomial of $A$, then $\operatorname{null}(A-\lambda_1 I)^{r_1} = r_1$

### Reputation (167)

 +5 Counterexample: $V \neq \ker(T) \oplus T(V)$ if $T^2 \neq T$ +30 Show that the only endomorphism $\phi$ of $\mathbb{Z}_7 \times \mathbb{Z}_7$ satisfying $\phi^5 = \text{id}$ is the identity. +10 $f, g: \mathbb{R} \to \mathbb{R}$ and $f(x+h) = f(x) + g(x)h + a(x,h)$ for $|a(x,h)| \leq Ch^3$. Show that $f$ is affine. +5 For $4\times 3$ matrix $M$, for any $3\times4$ matrix $N$, $\exists 0 \neq v \in \mathbb{C}^4$ such that $MNv = 0$.

 0 $f, g: \mathbb{R} \to \mathbb{R}$ and $f(x+h) = f(x) + g(x)h + a(x,h)$ for $|a(x,h)| \leq Ch^3$. Show that $f$ is affine. 0 Find a curve $\gamma$ satisfying $\int_\gamma y^3 \sin^2(x) \, dx - x^5 \cos^2(y) \, dy = 0$

### Tags (26)

 0 linear-algebra × 11 0 analysis × 3 0 matrices × 5 0 sequences-and-series × 3 0 abstract-algebra × 5 0 vector-spaces × 2 0 real-analysis × 4 0 minimal-polynomials × 2 0 jordan-normal-form × 3 0 group-theory × 2

### Account (1)

 Mathematics 167 rep 88 bronze badges