Linear Functions

In function notation \(f: \mathbf{R}^n \to \mathbf{R}\) means that \(f\) is a function that maps real \(n\)-vectors to real numbers, it is a scalar-valued function of \(n\)-vectors. In other words, if \(x\) is an \(n\)-vector, then \(f(x)\), which is a scalar, denotes the value of \(f\) at \(x\). That is, \(f\) returns a real number. Furthermore, we can interpret \(f\) as a function of \(n\) scalar arguments, as in …

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1% Rule

The 1% rule is a trading risk management technique that instructs the trader to never risk more than 1% of their account on any given trade. This article from the Balance tries to give some examples but it lost me at “By risking 1 percent of your account on a single trade, you can make a trade which gives you a 2-percent return on your account, even though the market only moved a fraction of a percent. …

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Juking For Linux

A very long time ago, I was a Windows user. I didn’t know any better. I even used Android. Those environments never sat well with me. Eventually, I “upgraded” into the Apple ecosystem, which although cohesive—as in, things just works—was too insular. The Microsoft, Apple, and Google triumvirate is not one that I’m fond of. Three times already, I tried to crossover into the Linux world. Three times I was thwarted by my own inexperience and incompetence. …

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Inner Product

The inner-product (also called dot-product) of two \(n\)-vectors is defined as the scalar \[a^Tb = a_1b_1 + a_2b_2 + \cdots + a_nb_n\] the sum of the products of corresponding entries. The \(T\) stands for transpose and we’ll tackle it another time. The dot product has other notations as well, such as \(\langle a, b \rangle\), \(\langle a | b \rangle\), \((a, b)\), \(a \cdot b\). As an example \[ \begin{bmatrix} -1 \\ 2 \\ 2 \end{bmatrix}^T \begin{bmatrix} 1 \\ 0 \\ -3 \end{bmatrix} = (-1)(1) + (2)(0) + (2)(-3) = -7 \] …

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Scalar-Vector Multiplication

Scalar multiplication (or scalar-vector multiplication) is an operation in which a vector is multiplied by a scalar (i.e., number), which is done by multiplying every element of the vector by the scalar. \[ (-2) \begin{bmatrix} 1 \\ 9 \\ 6 \end{bmatrix} = \begin{bmatrix} -2 \\ -18 \\ -12 \end{bmatrix} \] Similar notation is \(a/2\), where \(a\) is a vector, leading to \((1/2)a\). The scalar-vector product \((-1)a\) is simply \(-a\). The scalar-vector \(0a = 0\). …

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