Minimal Spanning Set vs Basis of a vector space

Question

I read the following in my textbook:

Find as small a set of vectors that span the row space of $A$ as you can. Such a set is called a minimal spanning set.

Is this terminology synonymous with the basis of the vector space? A basis is also made up of the largest set of linearly independent vectors that span a vector space.

Answer 1

Yes. The following three terms are equivalent (for a vector space!):

1. A linearly independent spanning set.
2. A minimal spanning set.
3. A maximal linearly independent set.

The first obviously implies the second and third. To see that 2. implies 1., suppose that if $\{x_1,\ldots,x_m\}$ is a minimal spanning set, but not a basis. Then, for some constants $\alpha_1,\ldots,\alpha_m$, not all zero, we have that

$$\displaystyle \alpha_1 x_1+\cdots+\alpha_m x_m=0$$

So, assume that $\alpha_1\ne 0$. Then,

$$x_1=\frac{-\alpha_2}{\alpha_1}x_2+\cdots+\frac{-\alpha_m}{\alpha_1}x_m$$

Thus, $\{x_2,\ldots,x_m\}$ is a spanning set (why?) and thus this contradicts minimality.

You and try to prove that 3 implies 1.