Had a go at learning some linear algebra through a great Khan Academy course today. Vectors seem to turn up in my work constantly and its something I’ve kind of gotten used to through exposure, but I don’t really have a solid grasp on the maths. Hopefully after this course I will!
All images are from the course, which can be found here.
What is a Vector?
A vector is a list of numbers that represents a direction and length in 2D or 3D space.
Adding two vectors together is like placing the second vector at the end of the first, and then drawing a line between the origin and the end of the second vector.
When we multiply a vector by a single number, we essentially stretch it out by that amount, or scale it. Therefore in linear algebra numbers tend to be called scalars.
The base vectors are the vectors that all others in our coordinate system are derived from. For example, the most common base vectors are î and ^ j (known as unit vectors) which are 1 in the x and y direction.
Any time we have a vector, it is basically a scaled version of the base vector.
Linear Combination and Span
The linear combination of two vectors is anytime a vector is scaled and added. The span of a set of vectors is all of the possible vectors that can be made using the linear combination. For example, with î and ^ j, the linear combination is made up of all possible 2D vectors, but if we were to use the vectors [1, 0] and [3,0] the linear combination would be all possible vectors on one line. If the vectors were both 0, the only linear combination would be 0.