Question

Check if the vector space of all polynomials is

infinite-dimensional (it is enough to find infinite number of
linearly independent vectors).

Answer 1

Yes, the vector space of all polynomials is infinite-dimensional because we can find an infinite number of linearly independent vectors.

Step-by-step explanation:

Yes, the vector space of all polynomials is infinite-dimensional. We can show this by finding an infinite number of linearly independent vectors.

For example, consider the set of polynomials {1, x, x^2, x^3, ...}. These polynomials are all linearly independent because no polynomial in the set can be written as a linear combination of the others.

Since we can keep adding higher powers of x to this set to create more linearly independent vectors, we have an infinite number of linearly independent vectors and the vector space of all polynomials is infinite-dimensional.