A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the other vectors in the set.

The dimension of a vector space is the number of vectors in a basis for the vector space.

The set of vectors {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is a basis for R^3 because it is linearly independent and spans R^3.

If a vector space has a finite basis, then it is called a finite-dimensional vector space.

The dimension of the vector space of all polynomials of degree less than or equal to n is n+1.

The set of vectors {(1, 2, 3), (4, 5, 6), (7, 8, 9)} is linearly dependent because the third vector can be expressed as a linear combination of the first two vectors.

If a set of vectors spans a vector space, then it is called a spanning set for the vector space.

The dimension of the vector space of all real-valued functions that are continuous on the interval [0, 1] is infinite.

The set of vectors {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is a basis for the vector space of all polynomials of degree less than or equal to 2 because it is linearly independent and spans the vector space.

If a set of vectors is linearly independent, then it is called a linearly independent set.

The dimension of the vector space of all real-valued functions that are differentiable on the interval [0, 1] is infinite.

The set of vectors {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} is a basis for the vector space of all polynomials of degree less than or equal to 3 because it is linearly independent and spans the vector space.

If a vector space has an infinite basis, then it is called an infinite-dimensional vector space.

The dimension of the vector space of all real-valued functions that are continuous on the interval [0, ∞) is infinite.

The set of vectors {(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)} is a basis for the vector space of all polynomials of degree less than or equal to 4 because it is linearly independent and spans the vector space.