A Vector Is a Movement
Say "vector" and you picture an arrow — but also a pair of numbers like (3, 2). Which one is the real vector? That probably tripped you up. The answer: both, and they're the same thing. A vector is a single movement: "go this far, in this direction." Draw that movement as an arrow and you get the picture; write down how far it goes along each axis and you get the number pair. Lock in just this view and vector addition, matrices, everything ahead falls into place. And this lesson reveals what those "î, ĵ" that lesson 5 quietly used actually are.
See a vector as a movement. (3, 2) means "go 3 steps right, 2 steps up." Drag the arrow around. The pair of numbers next to it changes along with it. Change the numbers instead and the arrow moves. The arrow and the number pair are just two ways of writing the same instruction. The picture is intuitive; the numbers are easy to compute with. Use whichever fits the moment. Moving freely between the two is the first step.
What does adding vectors do? It just chains the movements. Go along a first, then from there go along b, and your final stop is a + b. Lay the arrows tip-to-tail: start b at the end of a, and from the original start to the end of b is the sum. With numbers it's even easier — just add component by component. (3,2)+(1,4) is (4,6). Chain the movements: that's all vector addition is.
What happens when you multiply a vector by a number? The direction stays, only the distance changes. Times 2 goes twice as far, times 0.5 goes half — same direction. Multiply by a negative? Same length, flipped to the exact opposite way. Drag the scalar on the slider. The arrow grows, shrinks, and flips, all along the same line. Multiplying a vector by a number is "keep the direction, adjust the distance."
Now the key idea. Take two basic arrows: î, one step right, and ĵ, one step up. Just by scaling and adding these two, you can reach any point in the plane. (3, 2)? That's 3 of î plus 2 of ĵ, so 3î + 2ĵ. Drag the target point and watch it get assembled out of "so many î plus so many ĵ." Every vector in the plane is, in the end, just a recipe: some î plus some ĵ. When lesson 5 said a matrix only needs to know where î and ĵ go, this is exactly why — every point is a combination of î and ĵ.
So how far can you get with î and ĵ? The entire plane. That's what "2D space" means: every point you can reach by combining two different directions. What if your two directions both ran horizontally? No matter how you add and stretch them, you'd only ever make one horizontal line. You couldn't fill the plane. Toggle the two directions to be parallel and watch the plane shrink to a line. So space is "every point your directions can reach," and only when the directions are independent does the full plane open up. (This "parallel means flattened" connects to the determinant-zero of lesson 6.)