# Tagged Questions

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### Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
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### Finding loci of possible points satisfying vector simultaneous equations

I recently had an exam and a question came up which I was only partially able to answer. The question was the following: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be constant vectors in ...
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### Calculating Eigenvector Centrality & Betweenness Centrality formulas explained in simple terms

I'm currently working on a software application that has a function that analyses networks of people and the relationships between them. Two of the important variables we look at are Eigenvector ...
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### Elementwise normal to vector of unknowns and non-defined matrix multiplications

I wonder about statement (1). It is given that u is a column vector, A and B are constant, symmetric, square matrices of such size that the expressions on the left hand side of (1) are well defined. ...
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### Product rule type formula for $\nabla \cdot (M(x)v(x))$ where $M(x)$ is a matrix and $v(x)$ is a vector?

Let $M(x)$ be a $n\times n$ matrix with each element depending on $x$ a variable on $\mathbb{R}^n$. Let $v(x)$ be a vector. Is there a nice product rule formula for $\nabla \cdot (M(x)v(x))$?
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### Vector multiplication. Difference between scaler and dot product?

We just started a new class where the first topic is briefly talking about vectors and vector multiplication. All tying this into neural networks. I am a bit behind with the understanding of what the ...
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### Differentiating a non-linear functional with respect to a vector

I have the functional: $$F=v^T\times A \times v$$ Where $A$ is a function of $v$. The non-linear system of equations necessary to find $v$ is obtained doing: $$\frac{\partial F}{\partial v}=0$$ ...
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### What have Vectors and Matrices got to do with each other?

In my undergraduate course work I learnt Vectors (as in those in vector space with magnitude and direction) separately from Matrices - an $n \times m$ array of numbers. However, after sitting in for a ...
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### To show the inequality $\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$

Let $A\in$ $\mathbb{C}^{p\times q}$ with column $u_1,\ldots,u_q$ and rows $\vec{v_1},\ldots,\vec{v_p}$. show that $$\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$$ and ...
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### How would one use matrices to find a normal unit vector?

A recent class assignment involved finding a unit vector perpendicular to a plane, given two unit vectors to start with. The solution given involved using the cross product; I was wondering if such a ...
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### Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
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### Vector derivative with power of two in it

I want to compute the gradient of the following function with respect to $\beta$ $$L(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T \cdot \beta)^2$$ Where $\beta$, $y_i$ and $x_i$ are vectors. The ...
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### What does $Az=y$ and $A^Tx=0$ imply about the relationship between $x$ and $y$?

Suppose A is a $m \times n$ matrix and the vectors $x$ and $y$ are such that $Az=y$ for some vector $z$ and $A^T x=0$. Which one is correct? $x^Ty=0$ $||x||_2=||y||_2$ $||x||_2 < ||y||_2$ $x=ay$ ...
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### All the matrices that are orthogonal and have $q_1,q_2$

"Determine all the orthogonal matrices $Q=[q_1,q_2,q_3]$ that have as the first two columns the vectors $q_1=\frac{1}{\sqrt{6}}(-1,2,-1)^T, \ q_2=\frac{1}{\sqrt{3}}(1,1,1)^T$". I used the ...