Can a Transformation Matrix be one-to-one and not onto?

Thread starter suchara
Start date Aug 12, 2010
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Matrix Transformation Transformation matrix

In summary, a transformation matrix can be one-to-one and not onto, meaning that each input has a unique output but there may be elements in the output that are not mapped to. The difference between one-to-one and onto transformations is that the former does not have repeated outputs while the latter covers the entire output space. To determine if a transformation matrix is one-to-one and not onto, you can use the rank of the matrix. A transformation matrix can also be both one-to-one and onto, which is called a bijection. Real-world examples of one-to-one and onto transformations include mappings between two sets of data and bijective functions in mathematics.

Aug 12, 2010

suchara

Title says all..
A transformation matrix is one to one if its columns are linearly independant, meaning it has a pivot in each column
but what if it doesn't have a pivot in each row(i.e. not onto)? is it still one-to one?

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Aug 12, 2010

suchara

actually nvm,... I just realized it will map an Rⁿ vector onto R^m where m < n, but its still one-to-one

1. Can a transformation matrix be one-to-one and not onto?

Yes, a transformation matrix can be one-to-one and not onto. This means that every input has a unique output, but there may be elements in the output that are not mapped to by the transformation.

2. What is the difference between one-to-one and onto transformations?

A one-to-one transformation is a function where each input has a unique output, while an onto transformation is a function where every element in the output is mapped to by the transformation. In other words, a one-to-one transformation does not have any repeated outputs, while an onto transformation covers the entire output space.

3. How can you determine if a transformation matrix is one-to-one and not onto?

To determine if a transformation matrix is one-to-one and not onto, you can use the rank of the matrix. If the rank of the matrix is equal to the number of columns, then the transformation is onto. If the rank is equal to the number of rows, then the transformation is one-to-one. If the rank is less than both the number of rows and columns, then the transformation is one-to-one and not onto.

4. Can a transformation matrix be both one-to-one and onto?

Yes, a transformation matrix can be both one-to-one and onto. This type of transformation is called a bijection and it means that every input has a unique output and every element in the output is mapped to by the transformation.

5. What are some real-world examples of one-to-one and onto transformations?

A real-world example of a one-to-one and onto transformation is a mapping between two sets of data, such as matching students to their grades. Another example is a bijective function in mathematics, where every input has a unique output and every output is mapped to by the function.