I got problem in understanding the Im finding method

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In summary, the conversation is discussing different methods for finding the image of a linear transformation. The first method involves taking the column that does not have a zero in the zero row and extracting the vectors from it. The second method involves taking the "b" term to make the whole row zeros and extracting from it. The third method involves making the columns zeros. The conversation also highlights the confusion between the terms "kernel" and "image" in linear transformations. The correct method for finding the image is to solve the equation Au = <x+y, 0, 2x+z> rather than using the equation shown in the example.
  • #1
transgalactic
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i am puzzled about the method of finding the Im

in one axample i see the method of thaking the column withound the term of the zero
row
and exract the vectors from the column

the other way is taking the "b" term that make the whole row zeros
and extract from it

the third way i making the coulmns zeros

i am really confused about the method of finding Im
because there is 2 example the contaridct each other

http://img504.imageshack.us/my.php?image=img8271ip1.jpg
 
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  • #2
You seem to be consistently having trouble distinguishing between "kernel" and "image". If A is a linear transformation from vector space U to vector space V, then the kernel of A is a subspace of U and the image of A is a subspace of V.

In the problem before, where you were seeking the kernel, you found A0, a vector in V, when you should have been solving Au= 0 to get a vector in U.

Now, you appear to be solving the equation
[tex]\left[\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 0 & 0 \\ 2 & 0 & 1\end{array}\right]\left[\begin{array}{c} x \\ y \\ z \end{array}\right]= \left[\begin{array}{c} b_1 \\ b_2 \\ b_3 \end{array}\right][/tex]
when you should be calculating
[tex]\left[\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 0 & 0 \\ 2 & 0 & 1\end{array}\right]\left[\begin{array}{c} x \\ y \\ z \end{array}\right]= \left[\begin{array}{c} x+ y \\ 0 \\ 2x+ z\end{array}\right][/tex]

It is obvious that the the second component is always 0 and is easy to see that, since x, y, z are arbitrary, that the first and third components can be any numbers: the image is the set of vectors <x, 0, z> or, equivalently, the subspace with basis <1, 0, 0> and <0, 0, 1>.
 

Related to I got problem in understanding the Im finding method

1. What is the "Im finding" method?

The "Im finding" method is a problem-solving approach that involves breaking down complex issues into smaller, more manageable parts in order to understand and solve them more effectively. This method can be used in various fields, including science, mathematics, and engineering.

2. How does the "Im finding" method work?

The "Im finding" method involves several steps, including identifying the problem, gathering information, breaking down the problem into smaller parts, analyzing each part, and then putting the parts together to find a solution. This method encourages critical thinking and organized problem-solving.

3. Why is the "Im finding" method important in science?

The "Im finding" method is important in science because it allows scientists to approach complex problems in a systematic and logical manner. It helps to break down the problem into smaller, more manageable parts, making it easier to analyze and understand. This method also promotes collaboration and creativity in finding solutions.

4. How can I improve my understanding of the "Im finding" method?

To improve your understanding of the "Im finding" method, you can practice using it in various problem-solving situations. You can also seek guidance from experienced individuals, such as teachers or mentors, who can provide helpful tips and strategies. Additionally, there are many online resources and tutorials available to help you learn and improve your skills in this method.

5. In what situations is the "Im finding" method most useful?

The "Im finding" method is useful in any situation that involves a complex problem that needs to be solved in a structured and organized manner. It can be applied in various fields, such as science, technology, engineering, and mathematics, and is particularly useful in research, problem-solving, and decision-making processes.

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