# Existence and Uniqueness of Inverses

• I
• jolly_math

#### 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.

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

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.

No. A would be a column vector

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

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

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