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broegger
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how do you prove that if v is an element of V (a vector space), and if r is a scalar and if rv = 0, then either r = 0 or v = 0... it seems obvious, but i have no idea how to prove it...
Yes... if you pick your basis at (A, B, C) then the vector v becomes (a - A, b - B, c - C) and the zero vector is now (A, B, C).matt grime said:That proof requires you to pick a basis. If I pick a different basis, do you know that it still holds?
This statement is a mathematical proof that shows that if the product of two variables, r and v, is equal to 0, then either r or v must also be equal to 0. In other words, if one of the variables is 0, then the other must also be 0 in order for the product to equal 0.
Proving this statement is important because it demonstrates the relationship between multiplication and the concept of 0 as the identity element. It also lays the foundation for more complex mathematical proofs and equations involving variables and their products.
The "if...then" structure is a conditional statement that states the necessary condition (rv = 0) for the sufficient condition (r=0 or v=0) to be true. In this case, it is necessary for the product of r and v to be 0 in order for either r or v to be 0.
Sure, let's say r = 0 and v = 5. The product of these two variables, rv, would equal 0, satisfying the first part of the statement. Since r = 0, this also satisfies the second part of the statement, showing that either r=0 or v=0 in order for the product to equal 0.
This statement can be applied in various fields such as physics, engineering, and economics to solve equations and determine unknown variables. It can also be used to prove other mathematical theories and concepts. For example, it is a fundamental principle in linear algebra and matrix operations.