COORDINATE VECTOR Meaning and
Definition
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A coordinate vector refers to a vector that represents the coordinates or components of another vector with respect to a given basis. In linear algebra, a vector is often represented as an ordered list of numbers, which can be thought of as its coordinate vector.
The concept of a coordinate vector is essential for understanding vector spaces and linear transformations. Given a vector space V, a specific basis, usually denoted by {v₁, v₂, ..., vₙ}, can be chosen to represent V uniquely. The coordinate vector of a vector v with respect to this basis is then determined by the coefficients or scalars that multiply each basis vector to form the vector v itself.
For example, if V is a three-dimensional vector space and the basis is {e₁, e₂, e₃}, then any vector in V can be expressed as a linear combination of these basis vectors. The coordinate vector of a vector v would be represented as [x₁, x₂, x₃], where x₁, x₂, and x₃ are the scalar coefficients of e₁, e₂, and e₃ respectively.
Coordinate vectors are useful in various mathematical applications, such as solving systems of linear equations, performing matrix operations, and analyzing vector spaces. They allow for a concise and organized representation of vectors, enabling efficient computations and mathematical manipulations.
Etymology of COORDINATE VECTOR
The word "coordinate vector" has its etymology rooted in the broader concept of coordinates and vectors.
The term "coordinate" is derived from the Latin word "co-" meaning "together" and "ordinare" meaning "to order, arrange". It was first used in mathematics in the mid-17th century, primarily to describe a system of assigning numbers or quantities to definite points or objects in a space.
The word "vector" comes from the Latin word "vehere" meaning "to carry". It was introduced into mathematics in the late 19th century by J. Willard Gibbs and Oliver Heaviside as a way to describe a quantity that has both magnitude and direction.
When these two terms are combined into "coordinate vector", it refers to a vector that represents the coordinates of a point in a given coordinate system.
