How Do You Calculate the Preimage of a Vector Under a Linear Transformation?

In summary, the preimage is the original input of a mathematical function and is important in understanding the relationship between input and output, solving equations, and finding the inverse of a function. To find the preimage, you need to know the output and the function and use algebraic techniques. The preimage and inverse image are similar but refer to different types of functions. A function can have multiple preimages for a given output.
  • #1
Aristotle
169
1
Moved from a technical math section, so missing homework template
Consider the linear transformation T: R3 --> R3 /w T(v1,v2,v3)=(0, v1+v2, v2+v3)

What is the preimage of w=(0,2,5) ?I tried setting up the system of equations and got v1+v2= 2 and v2+v3=5 but after that I got kinda lost in how to find the individual solutions?
 
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  • #2
I think all you can really solve for sure is the interval between v1 and v3, I would put them in terms of v2.
You should end up with a description of the space that translates into (0,2,5).
 

What does "preimage" mean?

The preimage refers to the original input or starting point of a mathematical function. It is the value that is used as input to produce a specific output, or image.

Why is it important to find the preimage?

Finding the preimage is important because it helps us understand the relationship between the input and output of a function. It also allows us to solve equations and determine the inverse of a function.

How can I find the preimage of a function?

To find the preimage of a function, you need to know the output or image and the function itself. You can then use algebraic techniques to solve for the original input value.

What is the difference between a preimage and an inverse image?

The preimage and inverse image are similar concepts, but they refer to different types of functions. The preimage relates to the input and output of a single function, while the inverse image relates to the input and output of a function that is the inverse of the original function.

Can a function have multiple preimages?

Yes, a function can have multiple preimages for a given output or image. This is because different input values can produce the same output value in some cases. However, some functions may have a unique preimage for every output.

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