# Proving that a basis multiplied by an invertible matrix is a basis.

$$B = (v_1 , v_2,\ldots, v_n)$$ form a basis of a general vector space. Given $$P$$, an invertible matrix, prove that BP is a basis.

Here's my thought - we know that an invertible matrix is a product of multiplying elementary matrices. Meaning, for each $$v_i$$, multiplying it with $$P$$ will result in some elementary actions done to the vector. These actions are simply linear combinations of them - so, BP will be a basis with vectors that are all linear combinations of vectors in $$B$$, meaning, they will still span and be linear independent.

Is this direction correct? I'll formalize it, ofcourse, but wanted opinions. Thanks!

• In “Linear combinations of them” what does “them” refer to? If it’s the rows of $B$, then how do you know that all of these linear combinations of the rows are linearly independent?
– amd
Commented Jan 3, 2020 at 21:21

HINT: Consider a linear combination of $$BP$$ that is zero:
$$\sum_{k=1}^n \lambda_kPv_k = 0.$$
Because $$P$$ is invertible, you may apply $$P^{-1}$$ to both sides. Do you think you can finish?