Linnear Algebra Isomorphisms : prove that f + g is an isomorphism?

Instead, you should start with an arbitrary x+y, and then try to construct x and y that will make x+y rationalIn summary, the question is asking to prove or disprove whether the mapping f + g is an isomorphism from U to V, given that f and g are isomorphisms from U to V. Isomorphism is a stronger property than bijection, and requires the property of composition of functions to hold. To prove this, you must show that for h = (f + g), h(u + u') = h(u) + h(u') for all u and u' in U. This can be done by starting with an arbitrary x+y and constructing x and y to make x+y rational
  • #1
zeion
466
1

Homework Statement



Suppose f and g are isomorphisms from U to V. Prove of disprove each of the following statements:
a) The mapping f + g is an isomorphism from U to V.


Homework Equations





The Attempt at a Solution



I have no idea where to start.. do I need to show that f and g are 1-1 and onto?
Or do I go from something like f(u) + g(u) = (f+g)(u)?
Isn't that defined by a property of composition of function..?
 
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  • #2
Note that isomorphism is a stronger property than bijection (1-1 and onto).
If h is an isomorphism, it means that h(u + u') = h(u) + h(u') for all u and u' in U.
So you should how this for h = (f + g).
 
  • #3
There's a classic math problem: find two irrational numbers x and y such that x+y is rational. This often gives people a lot of trouble, because they spend all their effort trying to guess irrational x and y then checking if x+y is rational.
 

FAQ: Linnear Algebra Isomorphisms : prove that f + g is an isomorphism?

1. What is a linear algebra isomorphism?

A linear algebra isomorphism is a bijective linear transformation between two vector spaces that preserves the algebraic structure. In simpler terms, it is a one-to-one mapping between two vector spaces that preserves operations such as addition and scalar multiplication.

2. How do you prove that f + g is an isomorphism?

To prove that f + g is an isomorphism, we need to show that it is both a linear transformation and a bijective mapping. This can be done by showing that f + g preserves vector addition and scalar multiplication, and that it is both injective (one-to-one) and surjective (onto).

3. What does it mean for f + g to preserve vector addition and scalar multiplication?

If f + g is a linear transformation, it means that (f + g)(u + v) = (f + g)(u) + (f + g)(v) and (f + g)(c*u) = c*(f + g)(u), where u and v are vectors in the domain of f + g and c is a scalar. In other words, the operation of adding two vectors and multiplying a vector by a scalar is preserved under the transformation f + g.

4. Why is it important for f + g to be bijective?

If f + g is not bijective, it means that there are elements in the codomain that are not mapped to by any element in the domain. This would result in a loss of information and the transformation would not be reversible. Bijectivity ensures that the transformation is one-to-one and onto, preserving the full structure of the vector spaces.

5. What are some applications of linear algebra isomorphisms?

Linear algebra isomorphisms have many practical applications in fields such as computer science, engineering, and economics. They are used to represent and analyze data, to solve equations and systems of equations, and to understand the properties of vector spaces and transformations. They also play a crucial role in the development of machine learning algorithms and data analysis techniques.

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