Linear algebra ordered basis problem

In summary, the conversation discusses finding the beta and gamma coordinates of a given vector x, using a set of beta and gamma bases consisting of 3 vectors each. The concept of ordered basis is also mentioned. The conversation also shows the confusion of the speaker in understanding the concept and the solution provided by their instructor.
  • #1
priyathh
3
0
1. The problem statement

find the β coordinates ([x]β) and γ coordinates ([x]γ) of the vector [tex]x = \begin{pmatrix}-1\\-13\\
9\\
\end{pmatrix}
\in\mathbb R[/tex]

if [tex]{β= \begin{pmatrix}-1\\4\\
-2\\
\end{pmatrix},\begin{pmatrix}3\\-1\\
-2\\
\end{pmatrix},\begin{pmatrix}2\\-5\\
1\\
\end{pmatrix}}[/tex] and [tex]{γ= \begin{pmatrix}3\\-1\\
-2\\
\end{pmatrix},\begin{pmatrix}1\\4\\
-2\\
\end{pmatrix},\begin{pmatrix}2\\-5\\
1\\
\end{pmatrix}}[/tex]3. The Attempt at a Solution

i read my notes and as i understood it, an ordered basis is the linear combination that you use to obtain a specific vector in a vector space. I am not clear on the beta and gamma coordinates,
and i can't understand why the β and γ basis includes 3 vectors? I am thinking on the lines that x is obtained through a combination between the β coordinates and the given β , but that does not get me anywhere. please someone point me in the right direction! thank you

edit : ok i understand that β times the β coordinates would give the vector before the transformation, and γ
times the γ coordinates give the vector after transformation. but what exactly is x then? the vector before transformation or the vector after transformation?
 
Last edited:
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  • #2
ok i have finally managed to get an answer for this question but I am not sure of it at all, this is what i did

[β]x(a,b,c,d) = x where (a,b,c,d) is the beta coordinate of x

and i solved this equation and ended up getting some values for a b c and d which i wrote down as the β coordinates. is this wrong?
 
  • #3
our instructor solved this problem so anyone want to know the solution just let me know :)
 
  • #4
priyathh said:
our instructor solved this problem so anyone want to know the solution just let me know :)
I would like to know.
 
  • #5


I can provide some clarification on the concept of ordered basis and how it relates to the given problem. An ordered basis is a set of linearly independent vectors that span a vector space. In other words, any vector in the vector space can be expressed as a linear combination of the vectors in the ordered basis.

In this problem, the vector x is given and we are asked to find its β and γ coordinates. This means that we need to express x as a linear combination of the vectors in the β and γ bases. The β basis includes 3 vectors because it is a 3-dimensional vector space and we need 3 linearly independent vectors to span it. Similarly, the γ basis also includes 3 vectors.

To find the β coordinates, we need to solve the equation β1x1 + β2x2 + β3x3 = x, where β1, β2, and β3 are the coordinates of x in the β basis and x1, x2, and x3 are the components of x. Similarly, to find the γ coordinates, we need to solve the equation γ1x1 + γ2x2 + γ3x3 = x.

As for what x represents, it is simply a vector in the given vector space. It is neither the vector before nor after transformation, but rather a vector that can be expressed in terms of the given ordered bases. I hope this explanation helps you understand the concept better and guides you in solving the problem.
 

1. What is a linear algebra ordered basis problem?

A linear algebra ordered basis problem refers to a mathematical problem that involves finding a set of vectors that can be used as a basis for a vector space. The basis must be ordered, meaning that the vectors must be arranged in a specific order. This is necessary for operations such as matrix multiplication and solving systems of linear equations.

2. How do you determine if a set of vectors is a basis for a vector space?

In order for a set of vectors to be a basis for a vector space, it must satisfy two conditions: linear independence and span. Linear independence means that no vector in the set can be written as a linear combination of the other vectors, while span means that the set of vectors can be used to create any vector in the vector space through linear combinations.

3. What is the importance of an ordered basis in linear algebra?

An ordered basis is important in linear algebra because it provides a way to represent vectors and perform operations on them. It also allows us to easily define and manipulate vector spaces and determine properties of linear transformations. In addition, an ordered basis can be used to find solutions to systems of linear equations and calculate determinants and eigenvalues.

4. Can an ordered basis be non-unique?

Yes, an ordered basis can be non-unique. In fact, for a given vector space, there can be multiple valid ordered bases. However, any two ordered bases for the same vector space will have the same number of vectors, known as the dimension of the vector space.

5. How do you solve a linear algebra ordered basis problem?

To solve a linear algebra ordered basis problem, you must first identify the vector space and the given set of vectors. Then, you can use techniques such as Gaussian elimination or matrix operations to determine if the set of vectors is a basis. If the set is not a basis, you can manipulate the vectors to create a new set that satisfies the conditions for a basis. It is also important to keep the vectors ordered in a specific way, such as in a row or column matrix, for easier manipulation and calculation.

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