Linear Map Form: E x R to R [SOLVED]

In summary, a linear map is a mathematical function that takes elements from one vector space to another while preserving addition and scalar multiplication. It is represented by a matrix and can be used to transform elements from one vector space to another, typically from a space of real numbers. Linear maps are distinguished from nonlinear maps by their properties of linearity and are commonly used in scientific research for modeling and analyzing systems.
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quasar987
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[SOLVED] form of a linear map

Homework Statement


Say E is a linear space (not necessarily of finite dimension), and R is the real numbers. Say we have a (contiuous) linear form T from E x R to R. Can we say T is of such and such a form? Particularily, can we say that T=g1+g2 where g1:E-->R and g2:R-->R are linear forms?
 
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  • #2
Sure. Take (x,y) in ExR. Define g1(x,y)=T(x,0) and g2(x,y)=T(0,y). Is that what you meant? Doesn't that work?
 

FAQ: Linear Map Form: E x R to R [SOLVED]

1. What is a linear map?

A linear map is a mathematical function that takes elements from one vector space to another, while preserving the operations of addition and scalar multiplication. It is also known as a linear transformation or linear function.

2. What does "E x R to R" mean in the context of a linear map?

The notation "E x R to R" indicates that the linear map takes inputs from a vector space E and outputs to a vector space of real numbers, denoted by R. This means that the linear map transforms elements from E into real numbers.

3. How is a linear map represented?

A linear map is typically represented by a matrix, where each column corresponds to a different input variable and each row corresponds to a different output variable. The elements of the matrix represent the coefficients of the linear map.

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

A linear map follows the properties of linearity, such as preserving addition and scalar multiplication, while a nonlinear map does not. This means that the output of a linear map can be expressed as a linear combination of its inputs, while a nonlinear map may involve more complex relationships between inputs and outputs.

5. How are linear maps used in scientific research?

Linear maps are used in various fields of science, including physics, engineering, and economics. They are particularly useful in modeling and analyzing systems, as they can simplify complex relationships and provide a more efficient way to make predictions and solve problems.

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