The height of mountain of Baek-Du is 2770m. It can also be 2.77km. It is depending on the unit you want to use. The fact is, whatever you use 2770 or 2.77, both means Baek-Du's height, whatever you choose, the height never transmute. This is Applicable to vector. the vector can be (2,3), or (4,6) based on what basis is, inherent essence of the vector never be changed.
In order to translate 2770m into 2.77km, we have to divide 2770 into 1000, because 1000m equals 1km. We can do this with matrix product. On this point, the matrix can do converting meter value into kilometer value. When you need to think about the problem which uses kilometer unit, this matrix will do right thing.
This is what matrix do. They translate vectors from basis to basis, without transmuting vector, which makes to think about the problem much easier.