I was just wondering if the dimension of P4. Does that mean that all the vector space of all polynomials of degree less than or equal to four, has the dimension 4?
Also if a set of vectors (v1,v2,..,vn)is linear independent. And M is a linear map then M(v1),M(v2),…M(vn) is linearly independent
Best Answer
It doesn't make sense to talk about the 'dimension' of a single polynomial, since dimension is a concept attached to vector spaces. We can talk about the degree, which is the highest power of the independent variable that shows up in the polynomial; then the set of polynomials with coefficients in a given field (e.g. the real or complex numbers) forms a vector space. If we consider the vector space of polynomials of degree at most $n$, then it's a vector space of dimension $n$, with basis given by $\{1, x, x^2, ..., x^n\}$.
It's not true that the image of a linearly independent set is linearly independent; just choose $M = 0$, for instance. Counterexamples can be found whenever $M$ maps from a space to a space of lower dimension.
If $M$ is assumed to be injective, then the image of a linearly independent set is still linearly independent; the proof is instructive, and left to the reader.