# All Questions

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### linear algebra subspaces and polynomials

Let P3 be the vector space of all polynomials of degree less than or equal to 3: a. With p(x)=1-x^3 and q(x)=1+x-2*x^2, find the subspace of P3 spanned by p(x) and q(x) b. find the orthogonal ...
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### If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$.

Herstein 3.4.20: If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$. I don't understand why $\varphi$ needs ...
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### Volume of solid of rotation about x-axis

Rotation of the region bounded by x=2y^2-1, x=y^2 and x-axis about x-axis. I draw out the graph, and found intersection is at (1,1) and (-1,-1) So is it correct to continue doing by finding volume ...
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### Solvable by Radicals and Polynomials

Let $f(x)$ be an irreducible polynomial of degree $5$ over $\mathbb{Q}$ which is solvable by radicals. How do I show that its splitting field is contained in a field of the form $K(a)$, where $K$ is ...
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### One of these two operators is not invertible

I have a Hilbert space $H$ and a bounded self-adjoint operator $T$ on it with $||T||=1$. I've been trying to show that at least one of $I+T, I-T$ are not invertible, but I haven't been able to make ...
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### Relationship between cohomology and higher-homotopy

Let $M$ be a connected, compact, and orientable manifold ($H^3(M)\cong\mathbb{Z}$), and let $G$ be a simple Lie group satisfying $\pi_1(G)=\pi_2(G)=0$. Let $\pi_M(G)$ denote the set of homotopy ...
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### Applying the basic formula for binomial distribution

I'm pretty confused on how this works. In my class my teacher states that: Let $X$ be a random variable with $S_X = \{0,1\}$. $X$ follows a Bernoulli distribution if $P(X = x) = p^x(1-p)^{1-x}$ for ...
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### Re-initialize a 3D spline surface using different control points

I have a 3d spline surface is that is modeled with 65 points where the x,y,z position of each point is known. I want to keep the surface shape, but have different x,y positions for my control ...
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### Need Help Understanding Notation With Functions

Original picture: LaTeX approximation: $$f\color{blue}{\substack{(x)\\x\to\infty}}=\pm\sqrt{\frac{(x^2+x)^3}{\pi}}.$$ What does the notation highlighted in blue mean? I understand that ...
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### Let $R$ be a commutative ring with 1. If $R$ is a PID, then every prime ideal is either zero or maximal.

Let $R$ be a commutative ring with 1. If $R$ is a PID, then every prime ideal is either zero or maximal. My proof: Let $I = (p)$ be a non-zero prime ideal of $R$. Note that $p$ is prime. Since $R$ is ...
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### Tail field of random variables in $\mathbb{Z}$

Let $X_1, X_2, \ldots$ i. i. d. with values in $\mathbb{Z}$, define $S_0 := 0$, $S_n := X_1 + \cdots + X_n$ and $R_n := \{S_n = 0\}$ for $n \in \mathbb{N}$. Show that ...
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### Prove that the output of the function equals the determinant

Let $δ$ : $M_{2×2}$($F$) $→$ $F$ be a function with the following three properties. ($i$) $δ$ is a linear function of each row of the matrix when the other row is held fixed. ($ii$) If the two rows ...
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### Proving that the unit cube cannot be tripled (with straight edge and compass)

I would like to show the unit cube cannot be tripled using a straightedge and compass. I note that the side of a cube that has been tripled would have a side length of $\alpha=\sqrt[3]{3}$ But ...
I have the following and is solution but I don't understand how they arrived at the solution.Can someone direct me to how they arrived at this please ? $$\frac{z(z-1)(z+1)}{(z-3)^3}$$ The solution : ...