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What exactly does it mean when ColA "coincides with" some R^n, say, R^3? What is the difference between coinciding with and spanning the subspace?
R^n refers to a mathematical notation for a space of n dimensions, where n is a positive integer. It is commonly known as the n-dimensional Euclidean space and is a fundamental concept in linear algebra and geometry.
When two spaces coincide in R^3, it means that they occupy the same three-dimensional space and have the same coordinates, but may have different orientations or directions.
R^3 is a three-dimensional space, whereas R^2 is a two-dimensional space. This means that R^3 has three axes (x, y, z) and can represent three-dimensional objects, while R^2 has two axes (x, y) and can only represent two-dimensional objects.
A subspace in R^3 is a subset of the three-dimensional space R^3 that satisfies the four properties of a vector space: closure under addition and scalar multiplication, existence of a zero vector, and existence of additive inverses.
R^3 is used in various scientific fields, including physics, engineering, and computer graphics, to represent and analyze three-dimensional objects and phenomena. It is also used in data analysis and machine learning for visualizing complex datasets in three dimensions.