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⭐ Core Definition: Coordinate (vector space)

In mathematics, a set $B$ of elements of a vector space $V$ is called a basis (pl.: bases) if every element of $V$ can be written in a unique way as a finite linear combination of elements of $B$ . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to $B$ . The elements of a basis are called basis vectors.

Equivalently, a set $B$ is a basis if its elements are linearly independent and every element of $V$ is a linear combination of elements of $B$ . In other words, a basis is a linearly independent spanning set.

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Coordinate (vector space) in the context of Real coordinate plane

In mathematics, the real coordinate space or real coordinate n-space, of dimension $n$ , denoted $R$ or $\mathbb {R} ^{n}$ , is the set of all ordered $n$ -tuples of real numbers, that is the set of all sequences of $n$ real numbers, also known as coordinate vectors.Special cases are called the real line $R$ , the real coordinate plane $R$ , and the real coordinate three-dimensional space $R$ .With component-wise addition and scalar multiplication, it is a real vector space.

The coordinates over any basis of the elements of a real vector space form a real coordinate space of the same dimension as that of the vector space. Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension $n$ , $E$ (Euclidean line, $E$ ; Euclidean plane, $E$ ; Euclidean three-dimensional space, $E$ ) form a real coordinate space of dimension $n$ .

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Coordinate (vector space) in the context of "Real coordinate space"

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⭐ Core Definition: Coordinate (vector space)

Coordinate (vector space) in the context of Real coordinate plane