# Left invertible mapping left inverse of matrix

• fabbi007
In summary, the relation from R^2 to R^2 is left invertible, meaning there exists a left inverse, due to its one-to-one mapping. The mapping is also onto, meaning that the range of the mapping is equal to Y. The left inverse can be found by using the matrix A^-1, which can be applied to both sides of the equation y=Ax, resulting in A^-1y=A^-1Ax=x. In this case, the left and right inverse are the same since the matrix is square. It can be proven that if a mapping is both left and right invertible, there exists only one left inverse and one right inverse.
fabbi007

## Homework Statement

relation from R^2-->R^2 ( R is real line)

(y1) [0 1] (x1)
(y2) =[-1 1] (x2)

is this left invertible?

if so what is the left inverse?

y1,y2 are element in a 2by 1 matrix, same with x1, x2. the elemenst 0,1,-1,1 are in a 2x2 matrix. I did no know how to represent a matrix here.

## Homework Equations

A mapping is left invertible only if it is one-to-one

## The Attempt at a Solution

The above relation is one-to-one because x is getting mapped into y with a 2 dimensional relation into a 2 dimensional space. If we solve this we get y1=x2; y2=-x1+x2. Hence I concluded it is left invertible. Also the mapping is onto, since the Range of the mapping is yielding Y.

I am not sure how I can find the left inverse now. Any ideas

Is this what you're trying to represent? $$\left(\begin{array}{c}y_1 \\ y_2 \end{array}\right) = \left( \begin{array}{c c} 0 & 1 \\ -1 & 1 \end{array}\right) \left(\begin{array}{c}x_1 \\ x_2 \end{array}\right)$$

Click on equation above to see the LaTeX code I used.

If we write your matrix equation as y = Ax, you can apply the left inverse to both sides to get A-1y = A-1Ax = x. In other words, you can use the left inverse to solve for x.

BTW, you should be posting these questions in the Calculus & Beyond section. The questions you are asking are math questions, not engineering.

Mark44 said:
I
If we write your matrix equation as y = Ax, you can apply the left inverse to both sides to get A-1y = A-1Ax = x. In other words, you can use the left inverse to solve for x.

BTW, you should be posting these questions in the Calculus & Beyond section. The questions you are asking are math questions, not engineering.

Thanks for the latex code Mark. This is from my electrical engineering course. Hence I posted here. Also, the above mapping is right invertible because from the definition the range=Y. Would there be different left inverse and right inverse for a mapping if it is both one-to-one and onto? My guess from left inverse is for the mapping would be

$$\left( \begin{array}{c c} 1 & -1 \\ 1 & 0 \end{array}\right)$$

and since the mapping is right invertible too. the right inverse would be the same as left inverse, right?

$$\left( \begin{array}{c c} 1 & -1 \\ 1 & 0 \end{array}\right)$$

Yeah, I think they would both be the same in this problem. I'm not certain on this, but I think that if you're dealing with square matrices, the left-inverse and right-inverse will be the same. Where I think this comes into play is when you're dealing with non-square matrices.

Mark44 said:
Yeah, I think they would both be the same in this problem. I'm not certain on this, but I think that if you're dealing with square matrices, the left-inverse and right-inverse will be the same. Where I think this comes into play is when you're dealing with non-square matrices.

Thanks Mark. I get it now. It is indeed a square matrix and there is only one inverse. Also since from definitions if a mapping is both left and right invertible then it has an inverse, meaning only one inverse.

Mark44 said:
Yeah, I think they would both be the same in this problem. I'm not certain on this, but I think that if you're dealing with square matrices, the left-inverse and right-inverse will be the same. Where I think this comes into play is when you're dealing with non-square matrices.

How can you prove that if a mapping F:X->Y is both left and right invertible that there exists only one left inverse and one right inverse. I am trying to understand the theory, I could understand the example though. Can you give me a hint?

## 1. What is a left invertible mapping?

A left invertible mapping is a function that has a corresponding left inverse function. This means that if a function f maps an element x to an element y, then the left inverse function g maps y back to x. In other words, g(f(x)) = x.

## 2. How is a left invertible mapping related to matrices?

In linear algebra, a left invertible mapping is closely related to the concept of a left inverse of a matrix. A square matrix A is said to have a left inverse if there exists another matrix B such that BA = I, where I is the identity matrix. This means that multiplying A by its left inverse B results in the identity matrix, which is equivalent to the definition of a left inverse mapping.

## 3. What is the significance of having a left invertible mapping or left inverse of a matrix?

Having a left invertible mapping or left inverse of a matrix is significant because it allows us to "undo" the original mapping or matrix operation. This is especially useful in solving equations or finding solutions to systems of equations, as we can use the left inverse to isolate a variable or find the inverse function.

## 4. How can I determine if a matrix has a left inverse?

A square matrix A has a left inverse if and only if its columns are linearly independent. This means that the columns of A cannot be linearly combined to equal the zero vector. Another way to determine if a matrix has a left inverse is to check if its determinant is non-zero. If the determinant is non-zero, then the matrix has a left inverse.

## 5. Can a matrix have more than one left inverse?

No, a matrix can have at most one left inverse. This is because if a matrix has two left inverses, say B and C, then both BA = I and CA = I. But this would mean that B = BI = CAB = C, so the two left inverses must be the same. However, it is possible for a matrix to have no left inverse if its columns are linearly dependent or its determinant is zero.

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