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        Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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        votes
        0answers
        53 views

        What is the most efficient path for a robot without turning radius?

        I recently programmed a most efficient path for a robot going from $(x_1, y_1 , \theta_1)$ to $(x_2, y_2)$. The bot does a combination of turning and moving at any given point, based on the difference ...
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        votes
        0answers
        31 views

        What is n-dimensional DFT as a linear transformation matrix look like? How it can be expressed as a matrix multiplication?

        In the above picture, the Discrete Fourier Transform, for the case of 1-dimension is expressed as a matrix multiplication of the 1-D vector of length $N$ with a DFT matrix of dimension $N\times N$. I ...
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        votes
        0answers
        77 views

        Conjecture that relates matrix systems with some specific functions as solution sets

        what is written below is a conjecture that I posed , and I ask for a proof or a disproof of it .I have checked the conjecture from $n$=$1$ up to $n$=$10$ using Matlab, and all results were in ...
        7
        votes
        1answer
        113 views

        Making matrix positive semidefinite

        Let $A$ be a symmetric square matrix with all elements $A_{i, j} = \pm 1$, and additionally all diagonal elements of $A$ are $+1$. Let $f(A)$ be the smallest number of signs we need to change in $A$ ...
        4
        votes
        1answer
        121 views

        Nonlinear boolean functions

        Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements. I wonder if there is any known algorithm/construction that, given any $n\geq 1$, returns a boolean function $f:\mathbb{F}^n_2\rightarrow \...
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        votes
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        39 views

        how many marbles does the first bag contain? [on hold]

        Eighty five marbles are divided into two bags. The amount in the first bag is equal to 8/9 the amount of marbles in the second one. How many marbles does the first bag contain?
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        votes
        2answers
        132 views

        Degree of the variety of independent matrices of rank $\leq r$?

        Consider an $m$-by-$n$ matrix $A$ with entries in a field $k$; we can see $A$ as a point in the affine space $\mathbb{A}^{m n}$. The rank of $A$ will be $\leq r$ (where $r<\min(m,n)$) if and only ...
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        votes
        0answers
        43 views

        Can a rational symmetric matrix $A$ be diagonalized as $A = P \Lambda S$ for some $P, S $ in the general linear group?

        Let $A$ be a $3 \times 3$ symmetric matrix with rational entries. Does there exists a unique pair of matrices $P, S \in \text{GL}_3(\mathbb{Z} )$ depending on $A$ such that $A = P \Lambda S$, where $\...
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        vote
        0answers
        24 views

        Defining a notion of “volume of its lattice” for non-rational subspaces

        Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”: $$\...
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        vote
        1answer
        79 views

        How to compute inverse of sum of a unitary matrix and a full rank diagonal matrix?

        $C = A+D$, $A$ being a unitary matrix and $D$ a full rank diagonal matrix. Is there any easy way to compute $C^{-1}$ from $A^{-1}$ and $D$, if it exists? I am interested in this question, because my ...
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        18 views

        Can we find lower and upper bounds for a vector norm based on a compound inequality on vector elements?

        Consider the vector $x\in \mathbb{R}^n$. I want to choose $x$ such that $$\alpha\leq |x_i-x_j|\leq \beta$$ for all $i,j$ and $0<\alpha<\beta$. Do you think it is possible to find a condition ...
        2
        votes
        1answer
        78 views

        The effect of random projections on matrices

        Let $A\in\mathbb{R}^{n\times n}$ be a given normal matrix, i.e. $A^TA=AA^T$. Let $P_s\in\mathbb{R}^n$ be a random projection matrix to an $s$-dimensional subspace in $\mathbb{R}^n$. Suppose $\frac{A+...
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        votes
        0answers
        37 views

        Height distance problem [closed]

        A 70 foot pole stands vertically in a horizontal plane supported by three 490 foot wires, all attached to the top of the pole. Pulled and anchored to three equally spaced points in the plane. How many ...
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        votes
        1answer
        187 views

        Reference request: Oldest linear algebra books with exercises?

        Inspired by the recent success of my "soft question" here, I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way....
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        2answers
        224 views

        How do fractional tensor products work?

        [I asked and bountied this question on Math SE, where it got several upvotes and a comment suggesting it was research-level, but no answers. So I'm reposting here with slight edits, but please feel ...

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        山西福彩快乐十分钟
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