From the last chapter, we learned how to add and subtract vectors. We also know that how to expand or compress a vector using scalar multiplication. Now we can finally explain what is “Linear” in the word: linear algebra.
Say we have a set of many 2d vectors. Let’s call them:
I have changed the notation a bit, instead of putting a little arrow on the letter, I just bold the symbols.
Notice that every single one of them has to be in a 2d Cartesian plane because we cannot add 2d vector and 3d vector. Then, I pick a few of them, randomly expand or compress them by multiplying some numbers. Then add everything up. In the end, I would expect the vector to be another 2d vector, let’s call it \(\pmb{a}\):
the set of numbers \(c_1,c_2,c_3,...,c_n\) tells you how you expand or compress each vector. For example, I want to create a vector that is just half of the vector \(\pmb{v}_1\), then we can do:
If this is how you construct vector \(\pmb{a}\), we say that \(\pmb{a}\) is a Linear combination of \(\pmb{v}_1,\pmb{v}_2,\pmb{v}_3,\pmb{v}_4,...,\pmb{v}_n\). The word “Linear” just means you only used vector addition and scalar multiplication to create your vector. We will see why this is called “Linear” later. Spoil alert! it has something to do with how space changes after certain operations.
Now, what is so great about this linear combination? Well, a lot of ideas are coming from the linear combination, it is just we don’t call it that way before.
For example, you can get any point(vector) on a line by multiplying a vector. We can say it is just the linear combination of that vector. Similarly, you can get any point(vector) on a plane by “linear” combining two vectors. This is exactly how we construct the 2d Cartesian plane, by combining vector (1,0) and (0,1)!
Later on, you will see that the above examples will be explained using the new concepts in linear algebra.