Explaining Linear Independence

How can you tell whether two 2D vectors, if you scale and combine them many times over, will make a single line, or instead fill the entire 2D plane?

If you want a concrete example to try with the answers below, consider the two vectors [1, 2] and [3, 4].

Answer 1

Two 2D vectors will fill a plane if they are linearly independent, meaning one is not a scalar multiple of the other.

Conversely, if they are linearly dependent (one is a scalar multiple of the other), they will lie on a line.

Vectors are called "linearly independent" if the determinant of the matrix they form is non-zero.

First, pick one of the two vectors. It doesn't matter which one.

Then, multiply it by some number (pick any number).

This makes a new vector.

If we treat these vectors as points on a 2D plane, we can make a line with them.

We get a line by scaling a point by some multiple.

Point 1 on this line is one of the original two vectors (the one we chose first).

Point 2 is the new point we got by multiplying Point 1 by some (any) number.

Now, draw a line through those two points on a piece of paper.

Then, take the other original vector (the one we didn't scale earlier) and plot it as a point on the same graph.

Does that point land perfectly on that line we just drew?

Take the first of the two given vectors, scale it, then plot the line you just made, and check if the second given vector lies on that line.

Look at the vectors [2, 3] and [4, 6].

If you stare at them long enough, you'll notice that 4 is to 2 as 6 is to 3.

Now, consider the concrete example given at the top of the page.

How are these two instances different?