I'm having a bit of trouble understanding the concept of dimension of a span. I know that the dimension of a set of vectors is the number of vectors in the basis, but for the following questions I am not told that the set is a basis, nor that the set is linearly independent. How can I conclude the following:
- Suppose S is a collection of 93 vectors in $\mathbb{R(74)}$. Then, dim(span(S)) is greater than or equal to...
All I know that I can conclude from this question is that S is not a basis of $\mathbb{R(74)}$ since there are 93 vectors in S and therefore it is linearly dependent. I'm not sure if this can help me solve the question, however.
- Suppose S is a collection of 44 pairwise distinct vectors in $\mathbb{R(61)}$. Then dim(span(S)) is greater than or equal to...
I'm not sure what exactly it means that the vectors are "pairwise distinct". If anyone can provide a definition that would be greatly appreciated.
Thanks for any help!
pairwise distinct
" A collection of vectors, $\{v_1,v_2,v_3,\dots, v_n\}$ are considered "pairwise distinct" iff for every pair $i,j\in \{1,2,\dots,n\}$ if $i\neq j$ then $v_i\neq v_j$. I.e. no vector appears in the collection more than once. (note that in your problem though, simply being unequal does not mean much since $v\neq 2v$ when $v\neq 0$ but clearly $v$ and $2v$ are closely related) $\endgroup$