Undergrad Existence and Uniqueness of Inverses

[jolly_math]

Existence: Ax = b has at least 1 solution x for every b if and only if the columns span Rm. I don't understand why then A has a right inverse C such that AC = I, and why this is only possible if m≤n.

Uniqueness: Ax = b has at most 1 solution x for every b if and only if the columns are linearly independent. I don't understand why then A has a n x m left inverse B such that BA = I, and why this is only possible if m≥n.

Could anyone explain the logic behind this? Thank you.

 

[Office_Shredder]

Let's start with uniqueness. Suppose A has two columns that are linearly dependent. Can you find x that's not equal to 0 with Ax=0? Hint: it only has two nonzero elements

 

[jolly_math]

Let's start with uniqueness. Suppose A has two columns that are linearly dependent. Can you find x that's not equal to 0 with Ax=0? Hint: it only has two nonzero elements

No. A would be a column vector, and only x=0 would work. Why does this lead to the left inverse B such that BA = I, and why it is only possible if m≥n?

Thank you.

 

[Office_Shredder]

No. A would be a column vector

It's not. You might have misread my question, give it another look :)

 

[jolly_math]

I'm not sure what the answer is, could you explain the reasoning? Thank you.

 

[Office_Shredder]

Can you write out a 2x2 matrix which has two columns that are linearly dependent?