311.1.5.14 Use vectors to describe this set as a line in R^4

In summary, the conversation discussed examples of solving equations involving a "free" variable $x_4$, showing that it can be set to any value and still have a valid solution. The solution can be written as a vector with $x_4$ as the parameter, or as a sum of a constant vector and a scalar multiple of a parameter vector. It was emphasized that understanding basic definitions is important in these types of problems.
  • #1
karush
Gold Member
MHB
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Screenshot 2020-12-23 at 11.41.43 AM.png

ok, just now looking at some examples of how to do this $x_4$ is just a row with all zeros
 
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  • #2
ok don't see any takers on this one but here is a book example that might help, so we have...

$x_1+3x_4, \quad x_2=8+x_4, \quad x_3 =2-5x_4$ with $x_4$ free

$x=\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}
=\begin{bmatrix}3x_4\\8+x_4\\2-5x_4\\x_4\end{bmatrix}
=\begin{bmatrix}0\\8\\2\\0\end{bmatrix}= ...$

hopefully so far
Screenshot 2020-12-26 at 1.09.45 PM.png
 
  • #3
Frankly it looks to me like you have no idea what you are supposed to be doing!

Yes, since we are told that "\(\displaystyle x_1= 3x_4\), \(\displaystyle x_2= 8+ x_4\), and \(\displaystyle x_3= 2- 5x_4\) we have immediately that \(\displaystyle \begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}= \begin{bmatrix} 3x_4 \\ 8+ x_4 \\ 2- 5x_4 \\ x_4 \end{bmatrix}\).

But why then did you set \(\displaystyle x_4\) to 0?? The problem says that \(\displaystyle x_4\) is "free" which means that it can be any number. Saying that a number is "free" certainly does NOT mean that it is 0!

I would say that \(\displaystyle \begin{bmatrix} 3x_4 \\ 8+ x_4 \\ 2- 5x_4 \\ x_4 \end{bmatrix}\) is a perfectly good answer but some people might prefer to replace the "coordinate", \(\displaystyle x_4\) with the "parameter", t:
\(\displaystyle \begin{bmatrix} 3t \\ 8+ t \\ 2- 5t \\ t \end{bmatrix}\).

Some would prefer to write that as
\(\displaystyle \begin{bmatrix}0 \\ 8 \\ 2 \\ 0 \end{bmatrix}+\begin{bmatrix} 3 \\ 1 \\ -5 \\ 1 \end{bmatrix}t\).
 
Last edited:
  • #4
yes it is new material to me
 
  • #5
Then you need to start by learning the basic definitions!
 

1. What is the purpose of using vectors to describe a set as a line in R^4?

The purpose of using vectors is to represent the set as a line in four-dimensional space. This allows for a more efficient and accurate way of describing the set and its properties.

2. How do you use vectors to describe a set as a line in R^4?

To describe a set as a line in R^4, you need to use four-dimensional vectors that represent the coordinates of points on the line. These vectors can be manipulated to determine the direction and length of the line.

3. Can vectors be used to describe any set as a line in R^4?

No, vectors can only be used to describe sets that exist in four-dimensional space. If a set exists in a different number of dimensions, a different method must be used to describe it as a line.

4. What is the significance of using R^4 in vector descriptions?

R^4, or four-dimensional space, is used because it allows for a more comprehensive representation of the set. It also allows for more complex calculations and analysis to be performed on the set.

5. Are there any limitations to using vectors to describe a set as a line in R^4?

One limitation is that not all sets can be accurately represented as a line in four-dimensional space. Additionally, the use of vectors may not be the most efficient method for describing certain types of sets.

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