Finding Basis for Matrix Transformation: SBT = E_k

In summary, the conversation is about determining matrices S and T such that for matrix B, SBT is equal to a specific form with the identity matrix and zeroes. The rank of B is 2 and the vectors that solve Bx=0 are given. The person is struggling to express the columns of B as a linear combination of the given vectors and is looking for help on how to perform the transformation.
  • #1
Marin
193
0
Hi all!

I want to determine to matrices, S and T, so that for the matrix B,

[tex]B=\left(\begin{array}{cccc}-1&-2&-5&-3\\1&1&4&2\\4&-3&9&1\\2&-6&0&-4\end{array}\right)[/tex]

it´s true that:

[tex]SBT=\left(\begin{array}{cc}E_k&0\\0&0\end{array}\right)[/tex]

where E_k is the identity matrix of dimension k x k and k is the rank of B

I successfully calculated that rg(B)=2 and I also have the vectors that solve Bx=0:

[tex]v_1=\left(\begin{array}{c}-3\\-1\\1\\0\end{array}\right)[/tex]

and

[tex]v_1=\left(\begin{array}{c}-1\\-1\\0\\1\end{array}\right)[/tex]

I don´t know why, but I cannot express the columns of B as linear combination of v_1 and v_2. I checked my calculations twice...?


I´m pretty sure it´s a basis change problem, but instead of having the two basis I have the transformed matrix.

How is one supposed to do such transformations?



Any help will be much appreciated!

Thanks a lot in advance!
 
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  • #2
If the rank of B is 2 (image is two dimensional), then you can't write it in the form
[tex]SBT=\left(\begin{array}{cc}E_k&0\\0&0\end{array}\right)[/tex]
It would have to have rank 3 to be written that way.
 
Last edited by a moderator:
  • #3
no no, E_k is the identity matrix where k is 2=rgB so SBT is a 4 x 4 matrix in the top left corner of which we write the identity 2 x 2 matrix and zeroes elsewhere

but I ´m still searching for a way to perform this transformation...
 

1. What does SBT = E_k mean in the context of matrix transformation?

SBT = E_k represents a specific type of matrix transformation called the standard basis transformation. It means that the matrix being transformed (SBT) will be equal to the identity matrix (E_k), which is a square matrix with 1s on the main diagonal and 0s everywhere else.

2. Why is it important to find the basis for matrix transformation?

Finding the basis for matrix transformation is important because it allows us to understand the relationship between the input and output of a linear transformation. It also helps us to identify the key elements of the transformation and determine its properties.

3. How do you find the basis for matrix transformation?

To find the basis for matrix transformation, we need to first identify the standard basis vectors. These are the column vectors of the identity matrix. Then, we use these vectors to form a new matrix that represents the transformation. The columns of this new matrix will form the basis for the transformation.

4. Can there be more than one basis for a matrix transformation?

Yes, there can be more than one basis for a matrix transformation. This is because there can be multiple ways to represent a transformation using different sets of vectors. However, the standard basis transformation (SBT = E_k) is unique and is often used as a reference point for other basis transformations.

5. How does finding the basis for matrix transformation relate to eigenvalues and eigenvectors?

Finding the basis for matrix transformation is closely related to eigenvalues and eigenvectors. Eigenvectors are the special vectors that do not change direction during a transformation, and eigenvalues represent the scaling factor for these vectors. These eigenvectors and eigenvalues can be used to form the basis for a matrix transformation, providing a deeper understanding of the transformation and its properties.

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