Hallar BASE conociendo la MATRIZ de CAMBIO de BASE | Clase #4 | Álgebra para todos

Summary

In this lesson, we explore how to determine a second base of a vector space when given one base and the change of basis matrix. The instructor walks through an example of solving a system of linear equations, demonstrating the use of methods like substitution and reduction to find the unknown vectors. The lesson emphasizes understanding the relationship between vector bases and the importance of matrix operations in linear algebra. Finally, the instructor previews the next lesson on matrices associated with linear transformations, promising more in-depth coverage of the topic.

Q & A

What is the main objective of the lesson described in the transcript?

The main objective of the lesson is to demonstrate how to find the second base of a vector space, given the change of basis matrix and one of the bases.

What is the significance of the change of basis matrix in this context?

The change of basis matrix is important because it allows us to express the coordinates of vectors from one basis in terms of another basis, helping us to solve for the second base when one is given.

How is the change of basis matrix constructed?

The change of basis matrix is constructed by placing the coordinates of each vector from the given base (denoted as 'b') in terms of the new base (denoted as 'b prime') as columns of the matrix.

What does the vector 'b1' in the first column represent in the context of the change of basis matrix?

The vector 'b1' in the first column represents the coordinates of the vector 'b1' in the new base 'b prime.' The coordinates indicate how the vector is a linear combination of the vectors in the 'b prime' base.

How is the vector 'b1' expressed in the new base 'b prime'?

The vector 'b1' is expressed as a linear combination of the vectors in the 'b prime' base. For example, in the script, the vector 'b1' is written as 4 times the first vector of 'b prime' minus 1 time the second vector of 'b prime.'

What method is suggested for solving the system of equations derived from the change of basis problem?

The method suggested for solving the system of equations is the reduction method, which involves multiplying the equations by appropriate factors to eliminate variables and simplify the system.

What happens when the reduction method is applied to the system of equations in this lesson?

When the reduction method is applied, multiplying the first equation by 2 and adding it to the second equation allows the cancellation of the variable 'w,' leaving only the vector 'b' to solve for.

What is the result of solving for the vector 'b' using the reduction method?

The result of solving for the vector 'b' is the coordinates of 'b' in the new base 'b prime,' which are found to be (2, 1/5).

How is the vector 'w' determined once 'b' is found?

Once the vector 'b' is found, the equation involving 'w' is substituted with the values of 'b,' and then the vector 'w' is solved by performing simple arithmetic operations.

What is the final value of the vector 'w' in this example?

The final value of the vector 'w' is calculated to be (-2, 6/5).

What is the importance of practicing solving systems of equations involving vectors, as mentioned in the script?

Practicing solving systems of equations involving vectors is important because it helps students become comfortable with handling vector-based problems, particularly when the unknowns are vectors rather than scalar values.

What topic will be covered in the next lesson, according to the transcript?

In the next lesson, the focus will shift to the matrix associated with linear transformations and how it relates to the change of basis matrix, which will be further explored.