Search results
Results From The WOW.Com Content Network
238. A linear function fixes the origin, whereas an affine function need not do so. An affine function is the composition of a linear function with a translation, so while the linear part fixes the origin, the translation can map it somewhere else. Linear functions between vector spaces preserve the vector space structure (so in particular they ...
1. You know that a linear function satisfies the following property: f(a + b) = f(a) + f(b) f (a + b) = f (a) + f (b) and you want to determine whether a particular function g g is linear, so you just check whether this property holds. For example, we define the function g g as x ↦ 6x + 1 x ↦ 6 x + 1, thus:
Original Problem: Determine if the set of functions $$\{ y_1(x),y_2(x),y_3(x) \} = \{x^2, \sin x, \cos x \}$$ is linearly independent. I understand I have to use the Wronskian method, but how would it work for three functions with sine and cosine? Can someone help me give a brief overview of what I need to do and does the terms actually cancel?
4. An example that is close to the example you have of a linear transformation: f(x, y, z) = x + y f (x, y, z) = x + y. This is a linear functional on R3 R 3 or, more generally, F3 F 3 for any field F F. A much more interesting example of a linear functional is this: take as your vector space any space of nice functions on the interval [0, 1 ...
f(ax) = af(x) f (a x) = a f (x) where x x and y y are vectors and a a is a scalar. Roughly, this means that inputs are proportional to outputs and that the function is additive. We get the name 'linear' from the prototypical example of a linear function in one dimension: a straight line through the origin. However, linear functions can be more ...
1. This is very late, but hopefully it will help someone out. Rather than using a Sigmoid or Tanh function to transition from curve y1 to y2, try applying it to the gradient. Namely, aim for a smooth transition from the gradient of y1 to the gradient of y2. Example: Transition from y1(x) = x to y2(x) = 5. Make a sigmoid connecting the gradients ...
Linear is supposed to be f(cx1+bx2) = cf(x1) + bf(x2) where c and b are real numbers and x1 and x2 are elements of the domain/I/interval/whatever right? The definition of convex and concave use a and 1-a which only covers numbers in [0,1] so how are we extending this to all real numbers from just [0,1]? $\endgroup$
In calculus, a linear function is a polynomial function of the form f(x) = ax + b f (x) = a x + b. In linear algebra and functional analysis, a linear function is a linear map. (one of the properties that it satisfies is f(x + y) = f(x) + f(y) f (x + y) = f (x) + f (y), known as additivity) The difference between the two is that the latter ...
Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
An upper bound is given by the number of functions $\Bbb R\to\Bbb R$, which is also $2^{\frak c}$ - see here. If you do require continuity there are ${\frak c}=|\Bbb R|$-many continuous involutions. As an upper bound, there are $\frak c$ continuous real-valued functions - see here. As a lower bound, $\log_a\left(\frac{a^x+1}{a^x-1}\right)$ is a ...