Linear maps and composites HELP

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In summary, the conversation discusses a linear map T:V-->V in a finite dimensional vector space. It is mentioned that Ti+1 = TTi for all i >= 1 and suppose rank(T) = rank(T2). The statement "rank(T) = rank(T^2)" is considered in terms of the dimension of the image space of T and T^2. The question is asking to show that Ui is nonsingular for all i, which may involve relating the rank of a matrix to properties that imply non-singularity.
  • #1
stukbv
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I am told that T:V--> V is a linear map where V is a finite dimensional vector space.
i also know that Ti+1 = TTi for all i >= 1 Suppose rank(T) = rank(T2)

for i>= 1 Let Ui : Im(T)-->Im(T) be defined as the restriction of Ti to the subspace Im(T) of V. Show Ui is nonsingular for all i

I have no idea what this question is asking or how to attempt it!
 
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  • #2
consider the statement "rank(T) = rank(T^2)"

the rank of a matrix is equivalently:
- the number linearly independent row vectors
- the number linearly independent column vectors
- the dimension of the image space of T

consider it in terms of the third definition, meaning the image of T is the same dimension as the image of T^2
 
  • #3
I think they may be asking you to show that Ui is invertible (non-singular), with some given parameters. Try to relate the statements about the rank of the matrix to properties that imply non-singularity.
 

FAQ: Linear maps and composites HELP

1. What are linear maps and composites?

Linear maps and composites refer to mathematical functions that preserve the properties of linearity, such as scaling and addition. A linear map takes a vector from one vector space to another, while a composite is the composition of two or more linear maps.

2. How do linear maps and composites help in scientific research?

Linear maps and composites are essential tools in scientific research, particularly in fields such as physics, engineering, and economics. They allow for the modeling and analysis of complex systems and relationships, making it easier to understand and predict their behavior.

3. What is the difference between a linear map and a nonlinear map?

A linear map is a function that preserves linearity, meaning that the output is directly proportional to the input. In contrast, a nonlinear map does not follow this rule and may exhibit more complex relationships between the input and output.

4. How are linear maps and composites represented graphically?

Linear maps and composites are often represented graphically using vector diagrams or matrices. In a vector diagram, the input vector is represented as an arrow, and the output vector is the result of applying the linear map to the input vector. In a matrix representation, the linear map is expressed as a matrix, and the output vector is obtained by multiplying the input vector by the matrix.

5. How can I determine if a function is a linear map or a composite?

To determine if a function is a linear map, you can check if it satisfies the properties of additivity and homogeneity. Additivity means that the function's output when applied to the sum of two input vectors is equal to the sum of the function's outputs when applied to each input vector separately. Homogeneity means that the function's output when applied to a scaled input vector is equal to the scaled output. If these properties hold, the function is a linear map. A composite is a combination of two or more linear maps, so to determine if a function is a composite, you need to check if it can be written as the composition of two or more linear maps.

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