Linear Combinations

In we defined vector addition and scalar multiplication. These two operations combine nicely to give us a construction known as a linear combination, a construct that we will work with throughout this course.

1. 2x_1 + 3x_2 - x_3&= 0\\ x_1 + 2x_2 + x_3&= 3\\ x_1 + 3x_2 + 3x_3&= 7
2. S=\set{\colvector{1\\3\\-1},\,\colvector{2\\0\\4},\,\colvector{-1\\3\\-5}}
3. \begin{bmatrix} \leading{1}&3&0&6&0&9\\ 0&0&\leading{1}&-2&0&-8\\ 0&0&0&0&\leading{1}&3 \end{bmatrix}

So this definition takes an equal number of scalars and vectors, combines them using our two new operations (scalar multiplication and vector addition) and creates a single brand-new vector, of the same size as the original vectors. When a definition or theorem employs a linear combination, think about the nature of the objects that go into its creation (lists of scalars and vectors), and the type of object that results (a single vector). Computationally, a linear combination is pretty easy.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

We have written solutions to systems of equations as column vectors. For example has the solution $x_1 = -3,\,x_2 = 5,\,x_3 = 2$ which we write as \vect{x}=\colvector{x_1\\x_2\\x_3}=\colvector{-3\\5\\2}

Now, we will use column vectors and linear combinations to express all of the solutions to a linear system of equations in a compact and understandable way. First, here are two examples that will motivate our next theorem. This is a valuable technique, almost the equal of row-reducing a matrix, so be sure you get comfortable with it over the course of this section.

This is such an important and fundamental technique, we will do another example.

Did you think a few weeks ago that you could so quickly and easily list all the solutions to a linear system of 5 equations in 7 variables?

We will now formalize the last two (important) examples as a theorem. The statement of this theorem is a bit scary, and the proof is scarier. For now, be sure to convice yourself, by working through the examples and exercises, that the statement just describes the procedure of the two immediately previous examples.

Note that both halves of the proof of indicate that $\alpha_i=\vectorentry{\vect{x}}{f_i}$. In other words, the arbitrary scalars, $\alpha_i$, in the description of the set $S$ actually have more meaning they are the values of the free variables $\vectorentry{\vect{x}}{f_i}$, $1\leq i\leq n-r$. So we will often exploit this observation in our descriptions of solution sets.

formalizes what happened in the three steps of . The theorem will be useful in proving other theorems, and it it is useful since it tells us an exact procedure for simply describing an infinite solution set. We could program a computer to implement it, once we have the augmented matrix row-reduced and have checked that the system is consistent. By Knuth's definition, this completes our conversion of linear equation solving from art into science. Notice that it even applies (but is overkill) in the case of a unique solution. However, as a practical matter, I prefer the three-step process of when I need to describe an infinite solution set. So let us practice some more, but with a bigger example.

This technique is so important, that we will do one more example. However, an important distinction will be that this system is homogeneous.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

The next theorem tells us that in order to find all of the solutions to a linear system of equations, it is sufficient to find just one solution, and then find all of the solutions to the corresponding homogeneous system. This explains part of our interest in the null space, the set of all solutions to a homogeneous system.

After proving we commented (insufficiently) on the negation of one half of the theorem. Nonsingular coefficient matrices lead to unique solutions for every choice of the vector of constants. What does this say about singular matrices? A singular matrix $A$ has a nontrivial null space (). For a given vector of constants, $\vect{b}$, the system $\linearsystem{A}{\vect{b}}$ could be inconsistent, meaning there are no solutions. But if there is at least one solution ($\vect{w}$), then tells us there will be infinitely many solutions because of the role of the infinite null space for a singular matrix. So a system of equations with a singular coefficient matrix never has a unique solution. Notice that this is the contrapositive of the statement in . With a singular coefficient matrix, either there are no solutions, or infinitely many solutions, depending on the choice of the vector of constants ($\vect{b}$).

The ideas of this subsection will appear again in when we discuss pre-images of linear transformations ().

Condensed — the full section is in Beezer, A First Course in Linear Algebra.