LINEAR DEPENDENCE Meaning and
Definition
-
Linear dependence refers to a condition in linear algebra where a set of vectors can be expressed as a linear combination of other vectors in the same vector space. In other words, one vector in the set can be written as a sum of scalar multiples of the other vectors in the set. A set of vectors is said to be linearly dependent if there exists at least one vector in the set that can be expressed as a linear combination of the others.
More formally, given a set of vectors {v₁, v₂, ..., vₙ}, these vectors are linearly dependent if there exist scalars c₁, c₂, ..., cₙ, not all zero, such that c₁v₁ + c₂v₂ + ... + cₙvₙ = 0. The zero vector in this equation represents the additive identity in the vector space.
If a set of vectors is not linearly dependent, it is said to be linearly independent. A linearly independent set of vectors is one where no vector in the set can be expressed as a linear combination of the others. In other words, the only solution to the equation c₁v₁ + c₂v₂ + ... + cₙvₙ = 0 is when all the scalars c₁, c₂, ..., cₙ are zero.
Linear dependence is an important concept in linear algebra as it helps determine the dimension of a vector space and provides information about the existence of solutions to linear systems of equations.
Common Misspellings for LINEAR DEPENDENCE
- kinear dependence
- pinear dependence
- oinear dependence
- lunear dependence
- ljnear dependence
- lknear dependence
- lonear dependence
- l9near dependence
- l8near dependence
- libear dependence
- limear dependence
- lijear dependence
- lihear dependence
- linwar dependence
- linsar dependence
- lindar dependence
- linrar dependence
- lin4ar dependence
- lin3ar dependence
- linezr dependence
Etymology of LINEAR DEPENDENCE
The word "linear" is derived from the Latin word "linearis", meaning "belonging to a line". The term "dependence" comes from the Latin word "dependere", which means "to hang down".
In mathematics, "linear dependence" refers to the relationship between several mathematical objects (usually vectors or functions) that can be expressed as a linear combination of each other. The term emphasizes the concept that one object depends on or can be derived from others in a linear manner.
