Vectors stop being simple arrows the moment you understand that a few of them, combined with the right operations, can build entire planes or even three dimensional space. A linear combination of vectors is the operation that makes this possible, and it sits at the heart of how we describe colors on a screen, model physical systems and lay the groundwork for linear algebra.

What is a linear combination of vectors?

A linear combination is what you get when you scale a set of vectors and then add them together. You are using the two operations you already know, scalar multiplication and vector addition, working as a team.

For two vectors V1 and V2, any expression of the form A·V1 + B·V2 is a linear combination, where A and B are scalars you can tune freely [0:34]. Think of A and B as two knobs you turn before adding the vectors.

Why is it called linear? Because if you fix one scalar and vary the other, the result traces a straight line parallel to one of the vectors. Fix B and move A, you get a line parallel to V1. Fix A and move B, you get a line parallel to V2.

How do you compute a linear combination step by step?

Let me walk you through the example from the lesson. Take V1 = (1,2) and V2 = (2,1), with A = 2 and B = 1 [1:35].

1. Scale V1 by A: 2·V1 becomes a vector twice as long in the same direction.
2. Scale V2 by B: 1·V2 stays the same.
3. Add them using the parallelogram rule: slide the second vector to the tip of the first and draw the diagonal from the origin.

The diagonal is your result, and the closing side of the parallelogram lands parallel to V1, which is exactly why we use the word linear.

What is the span or generated space of a set of vectors?

If you can turn the A and B knobs freely, the question becomes: which points can you actually reach? That set of reachable points is called the span or generated space [3:13], and there are three possible outcomes.

• Two vectors pointing in different directions. You can triangulate any point on the plane. The span is the full 2D plane.
• Two collinear vectors, meaning they sit on the same line. No matter how you stretch or add them, you cannot escape that line. The span is a line, and the second vector is redundant.
• Both vectors are the zero vector. The only point you can generate is the origin. If only one is zero, the span collapses to a line through the other.

What does it mean for two vectors to be collinear? They point along the same direction, so one is just a scaled version of the other. Adding the second one gives you no new direction to explore.

How do the standard basis vectors build any vector in the plane?

The two most important vectors in the 2D plane are the standard basis vectors, written i hat and j hat [4:18].

• i hat has coordinates (1, 0), one unit along the X axis.
• j hat has coordinates (0, 1), one unit along the Y axis.

Any vector you draw on the plane is just a linear combination of these two. If V = (3, 2), then V = 3·i hat + 2·j hat. You move three units along X and two units along Y, and you land exactly on V [4:55].

Why are i hat and j hat so special?

Two properties make them the perfect building blocks: they are orthogonal and unit vectors [5:39].

• Orthogonal means they form a 90 degree angle, so neither one carries any component of the other.
• Unit means their length is exactly one, which keeps the math clean when you scale them.

These two ideas come back later in the course, but for now it is enough to know that orthogonality plus unit length make i hat and j hat the cleanest possible reference for the plane.

Where do you use linear combinations in real life?

You already use them every day, probably without noticing. The clearest example is the RGB color model [6:01]. Every color you see on a screen is a linear combination of three vectors: red, green and blue.

A shade of purple might be 0.5 of red plus 0.1 of green plus 0.8 of blue. Change the scalars and you change the color. That is a linear combination doing real work in your phone, your monitor and every video game you play.

What is the span of the RGB vectors? It is the full set of colors a screen can display. Three independent vectors generate a three dimensional color space, and each pixel picks one point inside it.

Practice exercise to test your understanding

Try this with two sets of vectors and share your graph in the comments [6:24]:

• Set A: U = (1, 1) and V = (-1, 1).
• Set B: U = (1, 1) and V = (1, 2).

Describe the span of each set. Are the vectors independent enough to cover the whole plane, or does one of them turn out to be redundant? That question of redundancy is exactly what linear independence will answer in the next class.