Precise definition of linear combination

In summary, a linear combination of a set of vectors is any sum with terms that are scalar multiples of those vectors. It does not matter what order they are added in, as long as they have the same coefficients. This is because vector addition is commutative. However, if even one of the coefficients is different, then it is considered a different linear combination. In other words, the concept of linear combination refers to the relation between a vector and a set of vectors, where the vector is equal to the sum of scalar multiples of the set of vectors.
  • #1
redjoker
5
0
i know that a linear combination of the vectors {v1,v2,...,vn} is any sum with terms that are scalar multiples of those vectors. But is a1v1 + a2v2 the same linear combination as a2v2 + a1v1? i know they evaluate to the same thing because vector addition is commutative but if i wanted to be precise would i say that it's the same linear combination or a different one only with the same set of coefficients? because if at least one of b1 and b2 was different from a1 and a2, then b1v1 + b2v2 is not considered to be the same linear combination even though they might be equal.
 
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  • #2
Hi redjoker! :smile:
redjoker said:
i know that a linear combination of the vectors {v1,v2,...,vn} is any sum with terms that are scalar multiples of those vectors. But is a1v1 + a2v2 the same linear combination as a2v2 + a1v1?

Yes!

And stop worrying :wink: … there really isn't a problem! :biggrin:
 
  • #3
The point is that linear combination refers to a relation between a vector, say x and a set of vectors say {xi}
A good definition is of Paul Halmos (in Finite-dimensional vector spaces):
We shall say, whenever x = Σiαixi, that x is a linear combination of {xi}

In other words, the phrase "x is the linear combination of..." is the synonym of "x is linearly dependent on...".
 

What is a linear combination?

A linear combination is a mathematical operation in which two or more quantities are multiplied by constants and then added together. It is often used to represent a relationship between two or more variables.

How is a linear combination expressed mathematically?

A linear combination is typically expressed in the form of a sum, where each term is a constant multiplied by a variable. For example, the linear combination 2x + 3y can be expressed as the sum of 2 times x and 3 times y.

What is the purpose of using a linear combination?

Linear combinations are used to analyze and describe relationships between variables in a mathematical and precise way. They can also be used to solve systems of equations and make predictions in various fields of science and engineering.

How does a linear combination differ from other mathematical operations?

A linear combination differs from other mathematical operations, such as addition or multiplication, in that it involves both multiplication and addition. It also allows for the use of variables and constants, making it a more flexible and powerful tool for solving equations and analyzing relationships.

In what fields of science is the concept of linear combination commonly used?

The concept of linear combination is commonly used in fields such as physics, chemistry, economics, and engineering, among others. It is an important tool for analyzing and describing relationships between variables in these fields and making predictions based on data.

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