Now we're going to start looking at vectors and matrices as more than just notational devices. They actually support algebra. Vectors. We're going to have two operations. The first is addition or equivalently subtraction. And in this case you need two vectors that have the same dimension, the same number of elements. And to add them, you just operate element-wise. It's a very natural thing to do. So if I were to add together these two three-dimensional vectors. The result is also three-dimensional. First element is two plus three. The next one is one plus 0, and the next one is negative one plus negative two. The other vector operation is called scalar multiplication. This term, scalar is new. And we'll be using it quite a bit. Really, it just means number as opposed to being a vector or a matrix. So to take a scalar times a vector, again, you just operate element-wise. If I want to multiply the scalar four times this vector, I just multiply each element by four. Really simple. We often package these two operations together into something called a linear combination. Linear combination I have n scalars, and I have the same number of vectors. And these vectors all have the same dimension, which I'll call M. So a linear combination of these is c1 times x1 plus c2 times x2, and so on all the way up to adding cn plus X n. So the result is an m-dimensional vector. The scalars in this case are known as the coefficients. This linear combination. Are we seeing this a lot both in linear algebra and differential equations? One reason that it's so useful is that it plays very nicely with linear functions. So if L is a linear function, it's not very hard to show using the properties of linearity. That L applied to a linear combination is equal to a linear combination of L applied to the individual vectors. This really unlocks a lot of things for us. One more thing to be aware of with linear combinations. Here's a system of three equations with three variables. I can rewrite it as a linear combination. So first, I write down the coefficients of x1 and I pull X1 outside the vector. It's a scalar multiplication. And that represents the first column in this system. Next, I have x2 times the coefficients of x2. That represents the second column in the equations. And finally, I have x3 times the coefficients of x three. So I make a linear combination out of those. And I set that equal to the terms on the right side, 0, negative 15. Now, if you start looking at equations, this first equation is actually represented in the first row of this linear combination. The second equation is represented by the second row, and so on. So I have two equivalent expressions for this linear system. This is an important thing for us to keep in mind is that a linear system of algebraic equations is equivalent to an equation about linear combinations.
I.6: Vector algebra
From Tobin Driscoll January 25, 2021
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