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woundedtiger4
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I assume that what you mean by that is that any 0-dimensional subspace must be the singleton set {(0,0,...,0)}. For this to be a good approach, you need a definition of "0-dimensional subspace" other than "contains (0,0,...,0) and nothing else". I would suggest "contains the additive identity and doesn't contain any linearly independent subsets". This takes us back to doing what Halls suggested, and also proving that every singleton subset of ##\mathbb R^n## other than {(0,0,...,0)} is linearly independent. This stuff about linear independence looks like an unnecessary complication. I would just do what Halls suggested.WWGD said:I am not sure I get the question either, but you can see at (0,0,..,0) as the only 0-dimensional subspace. Show any 0-dimensional subspace must coincide with (0,0,..,0).
Thanks a lotFredrik said:I assume that what you mean by that is that any 0-dimensional subspace must be the singleton set {(0,0,...,0)}. For this to be a good approach, you need a definition of "0-dimensional subspace" other than "contains (0,0,...,0) and nothing else". I would suggest "contains the additive identity and doesn't contain any linearly independent subsets". This takes us back to doing what Halls suggested, and also proving that every singleton subset of ##\mathbb R^n## other than {(0,0,...,0)} is linearly independent. This stuff about linear independence looks like an unnecessary complication. I would just do what Halls suggested.
The phrase "origin in R^n is the n-tuple 0=(0,0, .,0)" is a mathematical notation that means the point (0,0, .,0) is the starting point, or origin, in n-dimensional space. In other words, it is the point where all coordinates are equal to 0.
Showing that the origin in n-dimensional space is represented by the n-tuple 0=(0,0, .,0) is important because it is a fundamental concept in mathematics and physics. It allows us to define a reference point for measuring distances and angles, and it is essential for understanding vector operations and transformations.
The most common way to prove this statement is by using the definition of the origin and the properties of n-tuples. We can show that all the coordinates of the n-tuple 0=(0,0, .,0) are equal to 0, and therefore, it represents the origin in n-dimensional space.
No, the origin in n-dimensional space can only be represented by the n-tuple 0=(0,0, .,0). This is because the coordinates of the origin must be equal to 0, and any other n-tuple will have at least one non-zero coordinate.
Understanding the concept of the origin in n-dimensional space as the n-tuple 0=(0,0, .,0) has many practical applications. It is used in coordinate systems, vector calculus, and linear algebra, which are essential tools in various fields such as engineering, physics, and computer science.