How is dimension of a span of a set of vectors related to the number of vectors in the set?

Question

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:

1. 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.

2. 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!

Answer 1

Let $V$ be a vector space with finite dimension $\dim(V)=n$

Let $U=\{u_1,u_2,u_3,\dots,u_k\}$ be a collection of vectors from $V$

We have the following tight bounds:

$$0\leq \dim(\text{span}(U))\leq \min(n,k)$$

It could be as low as zero, for example in the case where all vectors in $U$ are the zero vector.

In the case that $k\leq n$ it could be as high as $k$ when all $k$ of the vectors are linearly independent and it could be no higher since if all $k$ of the vectors are linearly independent then $U$ would be a basis for $\text{span}(U)$, implying the dimension is exactly $k$.

In the case that $n\leq k$ it could be as high as $n$ when $\text{span}(U)=V$ and it could not be higher since that would imply we could find $n+1$ or more linearly independent vectors in $V$, contradicting that $\dim(V)=n$, i.e. that there is a basis of $n$ vectors in $V$ that spans the whole space.

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You ask for a definition of "pairwise distinct." A collection of objects $\{a_1,a_2,a_3,\dots,a_n\}$ are considered pairwise distinct iff for every $i,j\in\{1,2,3\dots,n\}$ we have $i\neq j\implies a_i\neq a_j$, or equivalently $a_i=a_j\implies i=j$. That is to say, no object appears in the collection more than once.

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For your specific problems:

In your specific problem, the first example could indeed have all vectors as the zero vector, so $0\leq \dim\text{span}(S)\leq 74$

$~$

In your second problem, we cannot have all be the zero vector since the vectors would no longer be pairwise distinct. We could however have our collection be distinct scalar multiples of the same nonzero vector. So, $1\leq \dim\text{span}(S)\leq 44$

Answer 2

For the first question:

Suppose S is a collection of 93 vectors in R(74). Then, dim(span(S)) is greater than or equal to..

We can say that dimension of the span(S) is somewhere between 0 and 74.

For the second question :

Suppose S is a collection of 44 pairwise distinct vectors in R(61). Then dim(span(S)) is greater than or equal to...

We can say that dimension of the span(S) is somewhere between 1 and 44.

The reason for 1 is that they are all distinct so there are at least one non-zero vector there.