Jeff
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 May 12 awarded Yearling Apr 19 answered Uniqueness in Matrix Multiplication Apr 8 comment If $B$ is a maximal linearly independent set in $V$ then $B$ is a basis for $V$ It is a complete proof. Think about it some more. Apr 8 comment If $B$ is a maximal linearly independent set in $V$ then $B$ is a basis for $V$ Yes. I don't understand your line of questioning. Are you unfamiliar with proofs by contradiction? Apr 7 comment If $B$ is a maximal linearly independent set in $V$ then $B$ is a basis for $V$ That is the contradiction. ($B \cup \{w\}$ is linearly independent by assumption.) Apr 7 answered If $B$ is a maximal linearly independent set in $V$ then $B$ is a basis for $V$ Mar 3 revised Solving an equation in charcateristic 2 in sage OR finding 3-torsion points of an elliptic curve over field with char 2 Said "2-torsion" when "3-torsion" was meant Mar 3 suggested approved edit on Solving an equation in charcateristic 2 in sage OR finding 3-torsion points of an elliptic curve over field with char 2 Feb 23 answered For what values of $k$ does this system of equations have a unique solution? Jan 16 answered Orthogonal Subspaces Jan 16 awarded Nice Question Dec 12 answered What is the basis of basis? Nov 29 comment Abstract Algebra: Field extensions Right. The norm will be fixed since the composition of two automorphisms $\sigma_i$ and $\sigma_j$ is another automorphism $\sigma_k$. Nov 28 answered Abstract Algebra: Field extensions Nov 25 comment Important topics to cover in an intro course to algebraic number theory? I can't upvote this answer enough. Oct 9 answered Differences between infinite-dimensional and finite-dimensional vector spaces Oct 3 comment Properties of Linear Transformations? Very often, the motivation for studying a map $T:A \to B$ between algebraic objects is a scenario where we understand $B$ very well, and we wish to understand $A$ better. In order to be able to transfer information from $B$ back to $A$, we need our map $T$ to preserve the relevant structures. Oct 3 comment Properties of Linear Transformations? Oh, and $\mathbf{R}^n$ is an example of a vector space. Oct 3 comment Properties of Linear Transformations? they are somehow less useful to study. If you study more mathematics, you will encounter many more examples of this sort of thing, and it will make more sense then. Oct 3 comment Properties of Linear Transformations? Jared's answer is focusing on the intuition. Mathematicians study many different types of objects; a vector space is an example of what might be called an algebraic object. A vector space is composed of things called vectors, and there are two operations you can carry out with these vectors: you can add them together, or you can scale them. So if we want to study maps (that is, functions) between two vector spaces, the best ones to consider will be the maps which respect those two operations. Thus, we get the definition of a linear transformation. Other maps between vector spaces exist, but