Example 6.79Let be the vector space of all differentiable functions. Consider the subspace W of given by W = span(e3, xe3, x2e3). Since the set B = {e3, xe3, x2e3} is linearly independent (why?), it is a basis
The ordinary vectors in ordinary space, or in "ordinary" n-dimensional space, form a vector space. Important vector spaces of functions are given by the continuous functions on an interval, the integrable functions, and the n times continuously differentiable functions. This chapter describes the ...
The elements of a vector space V are called vectors. The two closure properties require that both the operations of vector addition and scalar multiplication always produce an element of the vector space as a result. All sums indicated by “+” in properties (1) through (5) are vector sums...
We will generalize the examples of a plane and of ordinary space to and , which we then will generalize to the notion of a vector space. As we will see, a vector space is a set with operations of addition and scalar multiplication that satisfy natural algebraic properties. Then our next ...
We introduce a new class $$\mathcal {FV}(\Omega ,E)$$ of weighted spaces of functions on a set $$\Omega$$ with values in a locally convex Hausdorff space E
Suppose {eq}\, {\bf u}\, {/eq} and {eq}\, {\bf v}\, {/eq} are differentiable vector functions. Prove that: {eq}\qquad \dfrac{d}{dt}({\bf u}(t) + {\bf v}(t)) = {\bf u}'(t) + {\bf v}'(t) \, {/...
Explain the tangential and normal components of acceleration.We can combine some of the concepts discussed in Arc Length and Curvature with the acceleration vector to gain a deeper understanding of how this vector relates to motion in the plane and in space. Recall that the unit tangent vector ...
This is clearly a vector space over F. Up to this point the vector spaces have had recognisable components. In the next example this is not the case. Example 10: Diff(ℝ) is the set of all differentiable functions from the reals to the reals. The sum of two differentiable ...
Since all bases of a finite-dimensional vector space have the same number of elements, this number is defined to be the dimension of the space. Illustration ■ A two-dimensional vector space The space ℝ2 of all pairs of real numbers {a, b} is a two-dimensional vector space. The ...
If y = f1(x) and y = f2(x) are solutions of (1.3), then by the linearity of differentiation so are y = f1(x)± f2(x) and y = cf1(x), so the functions y = f(x) that represent solutions of (1.3) also form a vector space. ...