In this section we need to take a look at the equation of a line in R 3 R 3 . As we saw in the previous section the equation y = m x + b y = m x + b does not describe a line in R 3 R 3 , instead it describes a plane. This doesn’t mean however that we can’t write down an equation for a line in 3-D space. We’re just going to need a new way of writing down the equation of a curve. So, before we get into the equations of lines we first need to briefly look at vector functions. We’re going to take a more in depth look at vector functions later. At this point all that we need to worry about is notational issues and how they can be used to give the equation of a curve. The best way to get an idea of what a vector function is and what its graph looks like is to look at an example. So, consider the following vector function. → r ( t ) = ⟨ t , 1 ⟩ r → ( t ) = ⟨ t , 1 ⟩ A vector function is a function that takes one or more variables, one in this case, and returns a vector. Note as we
Example 6 Determine if the following set of vectors are linearly independent or linearly dependent. If they are linearly dependent find the relationship between them.
So, the first thing to do is to form and compute its determinant.
This matrix is non singular and so the vectors are linearly independent.
This matrix is non singular and so the vectors are linearly independent.
Example 7 Determine if the following set of vectors are linearly independent or linearly dependent. If they are linearly dependent find the relationship between them.
As with the last example first form and compute its determinant.
So, these vectors are linearly dependent. We now need to find the relationship between the vectors. This means that we need to find constants that will make true.
So, we need to solve the system
Here is the augmented matrix and the solution work for this system.
Now, we would like actual values for the constants so, if use we get the following solution,, and . The relationship is then.
So, these vectors are linearly dependent. We now need to find the relationship between the vectors. This means that we need to find constants that will make true.
So, we need to solve the system
Here is the augmented matrix and the solution work for this system.
Now, we would like actual values for the constants so, if use we get the following solution,, and . The relationship is then.
Calculus with Matrices
There really isn’t a whole lot to this other than to just make sure that we can deal with calculus with matrices.
First, to this point we’ve only looked at matrices with numbers as entries, but the entries in a matrix can be functions as well. So, we can look at matrices in the following form,
Now we can talk about differentiating and integrating a matrix of this form. To differentiate or integrate a matrix of this form all we do is differentiate or integrate the individual entries.
So, when we run across this kind of thing don’t get excited about it. Just differentiate or integrate as we normally would.
In this section we saw a very condensed set of topics from linear algebra. When we get back to differential equations many of these topics will show up occasionally and you will at least need to know what the words mean.
The main topic from linear algebra that you must know however if you are going to be able to solve systems of differential equations is the topic of the next section.
Comments
Post a Comment