Existence and Uniqueness of Inverses

In summary: Sure, let's say A = [1 2; 2 4]. Then the second column is just double the first column, so they are linearly dependent.
  • #1
jolly_math
51
5
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.
 
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  • #2
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
 
  • #3
Office_Shredder said:
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.
 
  • #4
jolly_math said:
No. A would be a column vector

It's not. You might have misread my question, give it another look :)
 
  • #5
I'm not sure what the answer is, could you explain the reasoning? Thank you.
 
  • #6
Can you write out a 2x2 matrix which has two columns that are linearly dependent?
 

1. What is the definition of an inverse?

An inverse is a mathematical operation that "undoes" another operation. In other words, it is the opposite of the original operation and when applied together, they cancel each other out.

2. What is the importance of existence and uniqueness of inverses?

The existence and uniqueness of inverses is crucial in mathematics because it allows us to solve equations and perform calculations with ease. It also ensures that every element in a set has a unique inverse, making the operations well-defined and consistent.

3. How do you prove the existence of an inverse?

To prove the existence of an inverse, we need to show that the operation is both one-to-one and onto. This means that for every input, there is only one corresponding output, and every output has a corresponding input. If these conditions are met, then the inverse exists.

4. Can an operation have more than one inverse?

No, an operation can only have one unique inverse. If an operation has more than one inverse, then it is not a valid operation. For example, addition and multiplication have unique inverses (subtraction and division, respectively), but subtraction and division do not have unique inverses.

5. What is the difference between a left inverse and a right inverse?

A left inverse is an element that, when applied to the left of an operation, yields the identity element. A right inverse, on the other hand, is an element that, when applied to the right of an operation, yields the identity element. In most cases, if an operation has a left inverse, it will also have a right inverse and vice versa.

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