Matrix invertibilty and equivalent statements

Hi!

I never took linear algebra, and I'm studying it on my own for the sake of learning (MIT opencourseware, online books, etc).

I'm currently working on that a matrix A is invertible, and I can show that an equivalent statement to this is that A is row equivalent to the n x n Identity matrix. Now, I'd like to show that the n x n identity matrix has n pivot positions. How would I get that started? I'm not looking for a formal proof, but an informal proof with justifications (doesn't have to be rigorous, just the sketch of the idea).

Thanks!

Re: Matrix invertibilty and equivalent statements

The real reason is that the identity matrix is, by definition, in rref with a leading 1 on the diagonal and zeroes everywhere else. Therefore there is a pivot position in each column (and row), and there are n columns (and row). There are also some geometric interpretations, but I'm pretty sure it all boils down to this property at the end anyways.

Re: Matrix invertibilty and equivalent statements

Thanks! That clarified it.

My next one to clarify is to show the following are equivalent (keeping in mind that A is an invertible n X n matrix) :

The columns of A span R^{n}.

The columns of A form a basis of R^{n}.

I'm still foggy on the idea of something "spanning" something else, as is the case of columns of A span R^{n}, and I don't know what it means that the columns of A form a basis of R^{n}. However, I do imagine that it is also true that the columns of A forming the basis of R^{n} implies that the columns of A span R^{n}. In other words, the first statement <==> the second statement.

So how are these two statements logically equivalent to one another?

Re: Matrix invertibilty and equivalent statements

Are you familiar with the invertible matrix theorem?

Re: Matrix invertibilty and equivalent statements

I'm familiarizing myself with it, but I noticed different texts show different paths to show the equivalence of those statements.

Re: Matrix invertibilty and equivalent statements

If I state that the columns of A are independent and they form a span of R^{n} therefore, they form a basis for R^{n}, is that enough or am I missing more of a justification?

I'm still foggy on the idea of span.

Re: Matrix invertibilty and equivalent statements

That is enough, assuming you know those two facts are true.

For your original question:

Quote:

The columns of A span Rn.

The columns of A form a basis of Rn.

The following applies to $\displaystyle n\times n$ matrices.

Columns of A span $\displaystyle \mathbb{R}^n \iff $ There are $\displaystyle n$ pivot rows in $\displaystyle A\iff $ There are $\displaystyle n$ pivot columns in $\displaystyle A \iff$ The columns of $\displaystyle A$ form a linearly independent set. Note that the middle equivalence about pivot rows and columns is not true in general for $\displaystyle m\times n$ matrices with $\displaystyle m\not= n$

Using what you have said in the preceding post, the columns of $\displaystyle A$ form a basis of $\displaystyle \mathbb{R}^n$. The reverse implication is trivial from the definition of basis.