# Tagged Questions

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### Let $V$ be the space of real sequences {${x_{1},x_{2},…}$} so that $\sum_{k=1}^\infty {x_{k}}^{2}$ converges. Prove that this space is not numerable

Let $V$ be the space of real sequences {${x_{1},x_{2},...}$} so that $\sum_{k=1}^\infty {x_{k}}^{2}$ converges. Prove that this space is not numerable: My attempt: I have already proved that this is ...
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### Acute angle between plane and line

Find the acute angle between: $x-y-3z=5$ and $x=2-t$ $y=2t$ $z=3t-1$ Here is how I proceed. I take the dot-product of the normal of the plane and the directional vector of the line. This gives me ...
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### I need help with linear transforms? Linear Algebra [closed]

In the question below, how was [T]ff found? I have tried but I can't understand how because I usually start from a given matrix with variables, but non is given here. website is here; ...
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### Proof Green's theorem $F(x,y)=(x-y)i+xj$

I was reading on Green's theorem and have appreciated the concept. Given a question, I think, I can solve it.But I came across a question that reads: Verify the Green's theorem for the vector given ...
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### Verifying The Divergence Theorem

Q: "Given the cylindrical region $x^2 + y^2\leq 1$ ,where $0\leq z \leq 1$, and the vector field $\underline{F} = 3x\underline{i} -5z\underline{k}$, verify the divergence theorem." For the ...
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### evaluating a flux integral

Question: "Region V, of unit volume, is bounded by the closed surface S. Given the vector field $\mathbf{F}=\langle 7x,2y,5z\rangle$, evaluate: $$\int_S \mathbf{F}\cdot\mathbf{dS}$$ I guessed that ...
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### Using Cylindrical Coordinates to Compute Curl

I am given a vector field $\vec{A} = A_\rho \space \hat{e_\rho} + A_\phi \space \hat{e_\phi} + A_z \space \hat{e_z}$, and I am supposed to use the unit vectors (provided below) in cylindrical ...
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### multivariable calculus question norm

Given vector space C([a,b],$\mathbb{R}$) of continuous functions of [a,b] in $\mathbb{R}.$ Prove that the function $\left \| f \right \|_{1}=\int_{a}^{b}\left | f(t) \right |dt$ is a norm. Also ...
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### multi-variate normal distribution distance from vector sub-space

let $X\sim {\cal N}(\mu,C)$ be a random variable obeying multi-variate normal distribution in $\mathbb{R}^n$ and $U \subset \mathbb{R}^n$ be a vector space with $\dim(U)=n-1$. What is the probability ...
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### How to find plane equation by line and plane that perpendicular to

Find an equation for the plane that is perpendicular to the plane 2x +2y=1 and passes through the line ...
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### Conditions for linear independence of extended vector systems

Assume $$g: R^n \times R^m \rightarrow R^n$$ $$h: R^n \times R^m \rightarrow R$$ $$(x,y) \in R^n \times R^m$$ I would like to show that the following vectors are linearly independent: ...
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### What is the difference between these two functions?

$$r(x) = \langle x, x^2-1 \rangle$$ $$f(x)=x^2-1$$ Their graph is the same, but one is called vector valued function while the other one is a regular one. I think I'll never get to understand this ...
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### what is the relationship between vector spaces and rings?

Can you show me an example to show how vector and scalar multiplication works with rings would be really helpful.
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### Finding vector length based on parallell and orthogonal vectors

Do anyone know a simple way of finding the length of vector a in my figure? The known values are $(x_0, y_0), (x_1, y_1), (x_2, y_2), (x_3, y_3)$. (If you look closely, you can see that $f$-vector is ...
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### Finding constant multiple of parallel vectors

Given two n-dimensional vectors that are parallel, is there any way to computationally find k such that $$\vec{v}_{1} = k\vec{v}_2$$ without prodding into the components?
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### How much can we “cheat” and use vector knowledge in complex analysis?

I'm an engineering-physics student taking a course in complex analysis, and it's a little frustrating, because I see all these connections to vector calculus over the reals (especially as applied to ...
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### Inner product of two complex vectors?

Given $A \in \mathbb R^{m \times n}$real matrix. If $\langle x,y\rangle =y^{*}x$ for all $x,y\in \mathbb C^{n \times 1}$, can someone help me find the relationship between the following two ...
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### Difference between Scalar field and a multivariable Function?

If a scalar field gives out a normal number for every orders pairs what's the difference between it and a function.
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### Line integral vs Arc Length

I am trying to understand when do to line integral and when to do arc length. So I know the formula for arc length varies based on $dx$ or $dy$ like so: $s=\int_a^b \sqrt{1+[f'(x)]^2} \, \mathrm{d} x$ ...
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### Integral with vector field in a circle.

Given $$F(x, y) = x^2\mathbf{i} + xy\mathbf{j}$$ $$x^2 + y^2 = 49$$ Find the work done by the force field on a particle that moves once around the circle oriented in the clockwise direction. I've ...
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### Parametric equation of line parallel to a plane

The parametric equation of the line is $$x=2t+1, y=3t-1,z=t+2$$ The plane it is parallel to is $$x-by+2bz = 6$$ My approach so far I know that i need to dot the equation of the normal with the ...
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### Basic arc length integration problem

How would you find the arc length of $r(t) = \langle\sqrt{t}, t,t^2\rangle$ for $1\le t\le 4$? This isn't a homework question, I'm just trying to understand how to properly solve a question such as ...
### inequality between entries of the vector and $l_2$ norm of the vector
Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. I am wondering for which vectors the following would be true: $$\|a\|_2^2\geq c \sum_{i\ne j,i,j}a_ia_j, \quad i,j=1, \ldots, n.$$ Here $c>0$ is ...