Elem lin algebra (Vector space question)

In summary, the conversation discusses the addition and scalar multiplication operations on a set of ordered pairs of real numbers, showing that the axiom -u + u = 0 does not hold true for these operations. This is because the additive inverse -v is not equal to the scalar multiple (-1)v in this case. This indicates that the given operations do not form a vector space.
  • #1
Miike012
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Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on u = (u1,u2) and v = (1,v2)

u + v = (u1+v1 ,u2+v2)

ku = ( 0 , ku2)

The book says that the axiom -u + u = 0 holds true for the given addition and scalar mult.

Which it obviously does not by the given scalar mult...
Hence: u + (-u) = (u1,u2) + (0,-u2) = (u1,0) ≠ 0.

Am I right or wrong?
 
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  • #2
What you have shown is that -v is NOT equal to (-1)v. "-v" is defined as "the additive inverse" of v and since the "addition" here is the usual "coordinate wise" addition, if v= <x, y> then -v= <-x, -y> while (-1)v= <0, -y>. Of course, since you can prove -v= (-1)v from the basic properties of a vector space, this says that this is not a vector space.
 
  • #3
-u usually means the additive inverse of (u1,u2). That would be (-u1,-u2). That's actually different from the 'scalar multiple' you've defined. (-1)*u=(0,-u2).
 

Related to Elem lin algebra (Vector space question)

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of elements (called vectors) and operations (such as addition and scalar multiplication) that satisfy specific axioms. These operations allow for the combination and manipulation of vectors in a way that follows certain rules.

2. What are the properties of a vector space?

The properties of a vector space include closure (the result of an operation on vectors is also a vector in the same space), associativity (the order of operations does not affect the result), commutativity (the order of vectors in an operation does not affect the result), and distributivity (scalar multiplication distributes over vector addition).

3. How is linear independence determined in a vector space?

Linear independence in a vector space is determined by whether or not a set of vectors can be expressed as a linear combination of other vectors in the space. If a set of vectors cannot be expressed in this way, then they are considered linearly independent. This means that none of the vectors can be created by combining the others, and each vector contributes unique information to the space.

4. What is the difference between a basis and a spanning set in a vector space?

A basis is a set of linearly independent vectors that can be used to express any vector in the space. A spanning set, on the other hand, is a set of vectors that can be used to create any vector in the space, but they may not necessarily be linearly independent. In other words, a basis is the minimum set of vectors needed to describe the entire space, while a spanning set may contain redundant or unnecessary vectors.

5. How are transformations represented in a vector space?

In a vector space, transformations are represented as linear operators. These are functions that take in a vector and output another vector in the same space. Linear operators can be represented by matrices, and certain properties of these matrices (such as their determinant and eigenvalues) provide important information about the transformation.

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