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== Examples ==
== Examples ==
# The space [[Euclidean space|<math>\mathbb R^n</math>]] of n-tuples of real numbers is a vector space, where to add two vectors we simply add the corresponding components. The case <math>n=2</math> is exactly the case of vectors in the plane discussed above.
# The space [[Euclidean space|<math>\mathbb R^n</math>]] of n-tuples of real numbers is a vector space, where to add two vectors we simply add the corresponding components. The case <math>n=2</math> is exactly the case of vectors in the plane discussed above.
# The set <math>\mathbb R[x]</math> of [[polynomial|polynomials]] with real coefficients is a vector space. If we add two polynomials together, we get another polynomial, and similarly, if we multiply a polynomial by a constant, we get another polynomial. Note that although it's also possible to multiply two polynomials and get another one, this is not part of the vector space structure: a vector space with a reasonable notion of multiplication of vectors is called an [[algebra]].
# The set <math>\mathbb R[x]</math> of [[polynomial|polynomials]] with real coefficients is a vector space. If we add two polynomials together, we get another polynomial, and similarly, if we multiply a polynomial by a constant, we get another polynomial. Note that although it's also possible to multiply two polynomials and get another one, this is not part of the vector space structure: a vector space with a reasonable notion of multiplication of vectors is called an [[algebra (mathematical structure)|algebra]].
# The set of polynomials of degree less than or equal to <math>n</math> (for any <math>n \geq 0</math>) is a vector space, for the same reason.
# The set of polynomials of degree less than or equal to <math>n</math> (for any <math>n \geq 0</math>) is a vector space, for the same reason.
# The set of all continuous functions on the real line is a vector space: the sum of two [[continuous|continuous functions]] is again continuous, as is the product of a continuous function with a constant.
# The set of all continuous functions on the real line is a vector space: the sum of two [[continuous|continuous functions]] is again continuous, as is the product of a continuous function with a constant.