[Math] Span and Dimension: A subspace

linear algebraproof-writingvector-spaces

If $A$ is finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$.

This is obviously true. Since $A$ is a finite set of linearly independent vectors and spans a subspace, $A$ is a basis for that subspace spanned by $A$ and thus by definition the dimension of a vector space is equal to the cardinality of any basis.

I would help with writing the above argument in a concise, precise manner with mathematical notation and other shorthand

Secondly in general what tips and/or advice you could give in general to make my arguments and proofs as efficient (time-wise) as possible.

Best Answer

Here are some tips that I follow when writing proofs.

  • Write in complete sentences including punctuation. (This seems contradictory since there are often so many symbols in math proofs. But symbols have exact meanings in words. For example, $\exists$ means "there exists". Anywhere you see $\exists$, in your mind you can replace that symbol with "there exists". In this way, math proofs should be paragraphs of complete sentences with punctuation.)
  • Write down the relevant definitions first. Often, the proof is just showing that the circumstances match the definitions.

I think you're trying to prove the statement: if $A$ is a finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$.

Here is one proof: The dimension of a vector subspace is the size of any of its bases. (Recall the theorem: all bases of a vector subspace have the same size.) A basis for a vector subspace $V$ is a set of linearly independent vectors that spans the subspace. We are given that $A$ is a set of linearly independent vectors. Therefore $\text{Span}(A)$ is a subspace, and its dimension is $|A|$ (the number of elements in $A$).

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