Effect of Basis Change on Vector Lengths in Different Vector Spaces
Does a change in the vector space of a matrix affect the length of vectors?
For instance, if in basis B, $$[v_1]_B = 3[v_2]_B$$, would it be possible in basis C for $$[v_1]_C ≠ 3[v_2]_C$$
for example:
$$[v_1]_C = 4[v_2]_C$$
given that B and C span the same vector space?
Basis change doesn't change length ratios. If $M$ is the matrix for basis change from $B$ to $C$, i.e. $[v]_C=M[v]_B$, then we have: $$[v_1]_C = M[v_1]_B = M\cdot 3[v_2]_B = 3\cdot M[v_2]_B=3[v_2]_C \ .$$
In general, if $\lambda \in \mathbb{R}$ and $[v_1]_B=\lambda [v_2]_B$, we have: $$[v_1]_C = M[v_1]_B = M\cdot \lambda[v_2]_B = \lambda \cdot M[v_2]_B=\lambda[v_2]_C \ .$$