14.1 find a vector v that will satisfy the system

In summary: . in summary, a basis for a vector space of dimension n must contain n vectors, any two of which imply the third, and the coefficients of a vector are determined by the multiplicity of the matrix corresponding to the vector.
  • #1
karush
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ok I think I got (a) and (b) on just observation

but (c) doesn't look like x,y,z will be intergers so ?
 
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  • #2
A basis for a vector space of dimension n has three properties:
1) The vectors are independent
2) The vectors span the space
3) There are n vectors in the space.
Further, any two of those imply the third.

The simplest way to answer the first question is that a basis for a three dimensional vector space, such as [tex]R^3[/tex], must contain three vectors, not two.

For the second, rather than using matrices, I would write
[tex]a\begin{bmatrix}1 \\ 1 \\ 1 \end{bmatrix}+ b\begin{bmatrix}1 \\ 2 \\ 3 \end{bmatrix}= \begin{bmatrix}a+ b \\ a+ 2b \\ a+ 3b\end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}[/tex]
so we must have a+ b= 0, a+ 2b= 0, and a+ 3b= 0. Subtracting the first equation from the second we have b= 0 and then a+ b= a+ 0= 0 so a= 0. The coefficients are a= b= 0 so the vectors are independent.
(Actually, two vectors are dependent if and only if one is a multiple of the other. Here, it is sufficient to observe that [tex]\begin{bmatrix}1 \\ 2 \\ 3 \end{bmatrix}[/tex] is not a multiple of [tex]\begin{bmatrix}1 \\ 1 \\ 1 \end{bmatrix}[/tex].)

There are, of course, infinitely many correct answers to
(c) Find a vector, v, such that [tex]\{\begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}, \begin{bmatrix}1 \\ 2 \\ 3 \end{bmatrix}, v\}[/tex] is a basis for [tex]R^3[/tex]. Since those are three vectors it sufficient to find v such that the three vectors are independent or such that they span [tex]R^3[/tex]. You appear to have chosen to find v such that the three vectors are independent but you haven't finished the problem. IF the three vectors were dependent then v would be a linear combination of the other two. In particular, taking the the coefficients to be 1, the sum of the two given vectors is [tex]\{\begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}+
\begin{bmatrix}1 \\ 2 \\ 3 \end{bmatrix}= \begin{bmatrix}2 \\ 3 \\ 4\end{bmatrix}
[/tex]
. To find a vector, v, that is NOT a linear combination, just change one of those components. Change, say, the "4" to "5": [tex]v= \begin{bmatrix}2 \\ 3 \\ 5 \end{bmatrix}[/tex]. The first two components are from "u+ v" while the third is not so v is not any linear combination.
(Since these are 3-vectors, you could also take v to be the "cross product" of the two given vectors. The cross product of two vectors is perpendicular to both so independent of them.)
 
  • #3
ok I can see that the multiplicity of the matrix's would be an easier way to check for linearity
but they seem to use the augmented matrix in the examples

thank you for the expanded explanation that was a great help
 

1. What is the meaning of "14.1 find a vector v that will satisfy the system"?

The phrase "14.1 find a vector v that will satisfy the system" is a mathematical statement that refers to finding a specific vector v that will make a given system of equations true. This means that when the vector v is substituted into the equations, the equations will have a solution.

2. What is a vector?

A vector is a mathematical quantity that has both magnitude (size) and direction. It is represented by an arrow pointing in a specific direction, and its length represents its magnitude. In this context, the vector v represents a set of values that will satisfy the given system of equations.

3. How do I find a vector v that satisfies the system?

To find a vector v that satisfies the system, you will need to use algebraic methods to solve the equations. This may involve manipulating the equations and using techniques such as substitution or elimination to find the values of the variables in the vector v that make the equations true.

4. Can there be more than one vector v that satisfies the system?

Yes, there can be multiple vectors v that satisfy the system. This is because a system of equations can have multiple solutions. In fact, there could be an infinite number of vectors v that satisfy the system, depending on the number of variables and equations in the system.

5. Why is finding a vector v that satisfies the system important?

Finding a vector v that satisfies the system is important because it allows us to solve the given equations and find the values of the variables that make them true. This can be useful in various applications, such as in physics, engineering, and economics, where systems of equations are commonly used to model real-world situations.

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