- #1
jwsiii
- 12
- 0
If you define a linear space as the set X of all functions from a nonempty set T into a field F with addition and scalar multiplication defined, would a subspace be the set X restricted to a certain subset of T?
matt grime said:The dimension, as a vector space of C(R) is not countably infinite.
rachmaninoff said:True, I just said continuous functions on a closed interval in R were countably infinite.
rachmaninoff said:True, I just said continuous functions on a closed interval in R were countably infinite.
Linear combination implies finite sum. Allowing infinite sums is beyond the algebraic sense, and moving into normed spaces, your countable set generates a dense subspace of C[0,1].
rachmaninoff said:True, I just said continuous functions on a closed interval in R were countably infinite.
Easy enough, [tex]\{ f(x)=c \, | \, c\in\mathbb{R}\}[/tex]. I'd meant to say their basis was countable, but that's false too.matt grime said:That's obviosuly false. Exercise construct uncountably many continuous functions on the interval [0,1]
Linear space, also known as vector space, is a mathematical concept used to describe a set of objects that can be added together and multiplied by a scalar. These objects can be anything from numbers to functions to physical quantities.
There are several key properties of linear space, including closure, associativity, commutativity, distributivity, and identity. Closure means that the result of adding or multiplying two objects in the space is also in the space. Associativity means that the order in which operations are performed does not affect the result. Commutativity means that the order of the objects being added or multiplied does not affect the result. Distributivity means that multiplication is distributive over addition. Identity means that there is an additive identity (0) and a multiplicative identity (1) in the space.
Linear space is a general concept that can be applied to any set of objects that satisfy the properties mentioned above. Euclidean space, on the other hand, specifically refers to a three-dimensional space with a defined origin and axes. Another key difference is that Euclidean space has a concept of distance and angles, while linear space does not.
Linear space has many practical applications, particularly in the fields of physics, engineering, and computer science. It is used to describe physical quantities such as force, velocity, and acceleration in physics. In engineering, it is used in areas such as control systems, signal processing, and optimization. In computer science, it is used in machine learning, image processing, and data analysis.
Linear algebra is the branch of mathematics that studies linear space and its properties. It provides the tools and techniques for solving problems involving linear space, such as finding solutions to systems of linear equations and performing transformations on vectors and matrices. Linear algebra is also used to model and solve real-world problems in various fields, making it an essential tool for scientists and engineers.