Change of Basis Homework: Solving System of Equations

In summary, the conversation discusses solving a system of equations to find the change of basis matrix from one set of bases to another. The equations are solved to determine the coefficients for each basis vector, which are then used to create the change of basis matrix.
  • #1
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Homework Statement

We are given 2 bases for V = [tex]\Re_{1 x 3}[/tex]. They are
[tex]\beta_{1}[/tex] = [tex] \begin{bmatrix} 2 & 3 & 2\end{bmatrix} [/tex]

[tex]\beta_{2}[/tex] = [tex] \begin{bmatrix} 7 & 10 & 6\end{bmatrix} [/tex]

[tex]\beta_{3}[/tex] = [tex] \begin{bmatrix} 6 & 10 & 7\end{bmatrix} [/tex]

and,

[tex]\delta_{1}[/tex] = [tex] \begin{bmatrix} 1 & 1 & 1\end{bmatrix} [/tex]

[tex]\delta_{2}[/tex] = [tex] \begin{bmatrix} 0 & 1 & 1\end{bmatrix} [/tex]

[tex]\delta_{3}[/tex] = [tex] \begin{bmatrix} 1 & 1 & 0\end{bmatrix} [/tex]

we are asked to find the [tex]\beta[/tex] to [tex]\delta[/tex] change of basis matrix.

The book says "by solving the relevant system of equations," you get

[tex]\beta_{1}[/tex] = [tex]\delta_{1}[/tex] + [tex]\delta_{2}[/tex] + [tex]\delta_{3}[/tex]

[tex]\beta_{2}[/tex] = 3[tex]\delta_{1}[/tex] + 3[tex]\delta_{2}[/tex] + 4[tex]\delta_{3}[/tex]

[tex]\beta_{3}[/tex] = 3[tex]\delta_{1}[/tex] + 4[tex]\delta_{2}[/tex] + 3[tex]\delta_{3}[/tex]


My question is WHAT system of equations did they solve to get the above?! I'm at a complete loss.

Homework Equations





The Attempt at a Solution


I know that for any vector [tex]\alpha[/tex], [tex]\alpha[/tex] = b1[tex]\beta_{1}[/tex] + b2[tex]\beta_{2}[/tex] + b3[tex]\beta_{3}[/tex] = d1[tex]\delta_{1}[/tex] + d2[tex]\delta_{2}[/tex] + d3[tex]\delta_{3}[/tex]. Where do I go from there?
 
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  • #2
They solved the 3 vector equations

[tex]\begin{align*}
\beta_1 &= d_{11} \delta_1 + d_{12} \delta_2 + d_{13} \delta 3 \\
\beta_2 &= d_{21} \delta_1 + d_{22} \delta_2 + d_{23} \delta 3 \\
\beta_3 &= d_{31} \delta_1 + d_{32} \delta_2 + d_{33} \delta 3
\end{align*}
[/tex]
 

1. What is a change of basis?

A change of basis is a mathematical process used to convert a set of coordinates or vectors from one set of basis vectors to another. It is typically used in linear algebra to simplify calculations or to solve systems of equations.

2. How does a change of basis help in solving systems of equations?

A change of basis can help in solving systems of equations by transforming the original system into a new system with simpler or more convenient basis vectors. This can make it easier to find solutions or to determine if a solution exists.

3. Can a change of basis affect the number of solutions to a system of equations?

Yes, a change of basis can affect the number of solutions to a system of equations. It is possible for a system to have a unique solution in one basis, but no solution or infinite solutions in another basis.

4. How do I know if a change of basis is necessary for solving a system of equations?

A change of basis is necessary when the given system of equations is difficult to solve or has a large number of variables. In these cases, transforming the system into a new basis can make it easier to solve or to determine if a solution exists.

5. What is the process for performing a change of basis on a system of equations?

The process for performing a change of basis on a system of equations involves finding a transformation matrix that converts the original basis vectors to the new basis vectors. This matrix is then used to transform the coefficients and constants in the system of equations, resulting in a new system that can be solved more easily.

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