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andytoh
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The question and my partial solution is in the link. Please help me finish it off. Thanks.
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No, that's not it. You are trying to prove that the columns are linearly independent.andytoh said:Thank you very much AKG. So the problem was that the hint was vaguely written. Had I known what they meant exactly, I could have found that eventually.
The columns of the nxn matrix in your last line are linearly independent and so the matrix is invertible. Left multiplying both sides of the equation by the inverse matrix results in the ai all being zero.
An uncountable infinite basis is a set of vectors that spans an infinite-dimensional vector space. This means that any vector in the space can be written as a linear combination of the basis vectors. The term "uncountable" refers to the fact that there are infinitely many basis vectors, and they cannot be counted or listed in a finite way.
A countable infinite basis is a set of vectors that spans a countably infinite-dimensional vector space. This means that the basis vectors can be listed or indexed in a countable way, such as the set of all positive integers. In contrast, an uncountable infinite basis has infinitely many basis vectors that cannot be listed in a countable way.
Uncountable infinite bases are important in mathematics because they allow us to study and understand infinite-dimensional vector spaces. These spaces have many applications in fields such as physics, engineering, and computer science. Additionally, uncountable infinite bases are closely related to other important mathematical concepts such as Hilbert spaces and Banach spaces.
Mathematicians use a concept called the Axiom of Choice to prove the existence of uncountable infinite bases. This axiom states that given any collection of non-empty sets, it is possible to choose one element from each set and form a new set. Using this axiom, mathematicians can construct uncountable infinite bases by choosing one basis vector from each dimension of the vector space.
No, not all vector spaces have an uncountable infinite basis. For a vector space to have an uncountable infinite basis, it must be infinite-dimensional and satisfy certain properties. For example, a finite-dimensional vector space cannot have an uncountable infinite basis. Additionally, some infinite-dimensional vector spaces may not have an uncountable infinite basis, depending on their structure and the choice of axioms used.