A question in different methods of a solution

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In summary, the conversation discusses a problem involving a linear transformation from 2 by 2 matrices to 2 by 2 matrices and finding the kernel and image of this transformation. The solution involves showing that a and b have no effect on the resultant matrix, and finding a basis for the kernel and image. It is pointed out that the 0 vector is always mapped to 0 and that the attempted solution is going in the wrong direction by looking at what 0 is mapped to instead of what gets mapped to 0.
  • #1
transgalactic
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i added the question and both solutions in the link

the solution that i was presented is totaly logical
but when i solved it in a standart way it gave me a different answer

http://img116.imageshack.us/my.php?image=img8272ib1.jpg
 
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  • #2
I hope it is no surprise that your book is correct. Basically the problem is this: you have a linear transformation from 2 by 2 matrices to 2 by two matrices defined by :
[tex]\left[\begin{array}{cc} a & b \\ c & d \end{array}\right][/tex] mapped to [tex]\left[\begin{array}{cc} 0 & c \\ 0 & d \end{array}\right][/tex]

Yes, since a and b "disappear" they play no part in determining the kernel. The only non-zero terms in the resultant matrix are c and d. It they are 0, then the result is the zero matrix. What that means, then, is that a and b can be anything at all as long as c= 0 and d= 0. Taking a= 1, b= 0 gives the matrix
[tex]\left[\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right][/tex]
and taking a= 0, b= 1 gives the matrix
[tex]\left[\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right][/tex]
Those two matrices form a basis for the kernel.

I don't think it really makes sense to talk about "a and b don't matter in determining the Image". a and b are only in the matrix the transformation is applied to- not in the resultant matrix. Obviously, every matrix in the image have first column 0. The second column can be anything. A basis for the image is
[tex]\left[\begin{array}{cc} 0 & 1 \\ 0 & 1 \end{array}\right][/tex]
and
[tex]\left[\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}\right][/tex]

In your "totally logical" solution, you are applying a matrix to the 0 vector. The "kernel" of a transformation is the set of vectors that are mapped to the 0 vector, not that the 0 vector is mapped to. You are going the wrong way.

(Any linear transformation maps the 0 vector to the 0 vector.)
 
  • #3
i can't understand
"not that the 0 vector is mapped to"

can you describe it a different words my mistake??
 
  • #4
If Ax= y, then "A maps x to y".

x is in the kernel of A if and only if Ax= 0. That is, if "A maps x to 0".

For any linear transformation, A, A0= 0. That is "A always maps 0 to 0".

In your attempted solution you construct an 8 by 8 matrix and apply it to the 0 vector. Of course, you get the 0 vector. That has nothing to do with a "kernel" because that will always be true. It going the wrong way: you are seeing what 0 is changed into (mapped to) rather that seeing what gets changed to 0.
 
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Related to A question in different methods of a solution

1. What is the scientific method?

The scientific method is a systematic approach used by scientists to conduct experiments and gather evidence in order to answer questions and solve problems in the natural world. It involves making observations, forming a hypothesis, designing and conducting experiments, analyzing data, and drawing conclusions.

2. How is the scientific method used in finding a solution?

The scientific method is used in finding a solution by providing a systematic and objective approach to problem-solving. It helps identify the cause of a problem, develop a hypothesis, and test it through experiments. The results of these experiments can then be used to form a solution or determine the best course of action.

3. What are the different methods used in the scientific method?

There are several different methods used in the scientific method, including observation, experimentation, data analysis, and forming a hypothesis. These methods are used in a cyclical process, with each step informing the next, until a solution is found.

4. How does the scientific method differ from other problem-solving methods?

The scientific method differs from other problem-solving methods in that it is systematic, objective, and based on evidence. Other methods may rely more on intuition or personal experiences, whereas the scientific method follows a specific process and relies on data to support conclusions.

5. Can the scientific method be applied to all problems?

The scientific method can be applied to many different types of problems, particularly those related to the natural world. However, it may not be applicable to all problems, such as those involving ethical or moral considerations. In these cases, other problem-solving methods may be more appropriate.

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