Linear Combination
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Consider a random pair of vectors, and :
Any time you’re scaling two vectors and adding them like this, it’s called a “linear combination” of those two vectors.
When you multiply a scalar by a vector, it changes the magnitude of that vector.
Multiplying every real number by the vector produces an infinite line that passes through the origin and the point defined by the vector.
So a linear combination of two vectors is a method of combining these two lines.
For most pairs of vectors, if you let both scalars range freely and consider every possible vector you could get, you will be able to reach every possible point on the plane.
Every two-dimensional vector is within your grasp.
However, if your two original vectors happen to line up, the lines produced by the scalar multiplication will be the same line, so adding them together can't yield a vector outside of that line.
There’s a third possibility too: Both your vectors could be the 0 vector, in which case you’ll just be stuck at the origin.
The set of all possible vectors you can reach with linear combinations of a given pair of vectors is called the “span” of those two vectors.
Restating earlier, the span of most pairs of 2D vectors is all vectors in 2D space, but when they line up, their span is all vectors whose tip sit on a certain line.
The span of two vectors is basically a way of asking what are all the possible vectors you can reach using these two by only using those fundamental operations of vector addition and scalar multiplication.
If two vectors are collinear, they are linearly dependent.
one of them is redundant and do not add anything to the span.
The technical definition for the “basis” of a space is a set of linearly independent vectors that span that space, given how I described a basis earlier, and given your understanding of the words “span” and “linearly independent”, why does this definition make sense?
Earlier we learned that any pair of vectors could form a new basis as long as they didn't line up. If pair of vectors are linearly independent, their linear combination can span the entire 2D plane, meaning they can form the basis for the plane.
