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Math Help - Finding the N(F) and V(f) the kernel and a values of a matrix, Questions

  1. #1
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    Finding the N(F) and V(f) the kernel and a values of a matrix, Questions

    Hello, I would like you to read the attached image where I have presented the entire problem and my two questions
    Finding the N(F) and V(f) the kernel and a values of a matrix, Questions-vf.png
    *
    Please see the attached file

    To clarify question 2 a bit more: How can i judge by the one parametric value of t:s which dim to take?
    How do i know its a line,vectors parallel to a plane, or vectors in a room?

    Is there any analytical way to check this?

    Thanks
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Finding the N(F) and V(f) the kernel and a values of a matrix, Questions

    Have you tried to solve the system? If you do you'll obtain one zero row that means you have to choose one parameter. Solving the system:
    \left \{ \begin{array}{ll} x_1+2x_3=0 \\ x_2-3x_3=0 \end{array} \right.
    Choose x_3=t so we obtain:
    \left \{ \begin{array}{ll} x_1=-2t \\ x_2=3t \\x_3=t \end{array} \right.
    The solution set is:
    V=\left \{ \left ( \begin{array}{ccc} -2t \\ 3t \\ t \end{array} \right) | t \in \mathbb{R} \right \}
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  3. #3
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    Re: Finding the N(F) and V(f) the kernel and a values of a matrix, Questions

    Actually I am studying linear alg on my own, i have never been to a lecture on linear algebra, thats why i have limited knowledge, and I have an exam that I am trying to pass very soon, i dont have a particular book either, im trying to read through some free books online sometimes, and i borrowed one book from the library where I saw the dim values., however they did not explain how to set the dim values.

    * Thanks for your answer Siron, but could you specify what a one zero row looks like?
    * I dont understand why you pick x3 = t, could you pick anyone to be t?


    Could someone answer that second question? How to know whether something is a dim vl =1, dim vpi = 2 , dim V = 3
    In The exam problem that i presented to you above which was studying they got dim v(F) = 2. I still dont get it though.
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  4. #4
    MHF Contributor Siron's Avatar
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    Re: Finding the N(F) and V(f) the kernel and a values of a matrix, Questions

    We are given the system:
    \left \{\begin{array}{lll} 2x_1+4x_3=0 \\ x_1+x_2-x_3=0 \\ -x_1+3x_2-11x_3=0\end{array} \right.

    We can solve the system if we apply row operations on the following matrix:
    \left ( \begin{array}{ccc} 2&0&4 \\ 1&1&-1 \\ -1&3&-11 \end{array} \right)
    =\left ( \begin{array}{ccc} 1&0&2 \\ 1&1&-1 \\ -1&3&-11 \end{array} \right)=\left ( \begin{array}{ccc} 1&0&2 \\ 0&1&-3 \\ 0&3&-9 \end{array} \right)=\left ( \begin{array}{ccc} 1&0&2 \\ 0&1&-3 \\ 0&1&-3 \end{array} \right)=\left ( \begin{array}{ccc} 1&0&2 \\ 0&1&-3 \\ 0&0&0 \end{array} \right)

    Therefore we have reduced the given system to:
    \left \{ \begin{array}{ll} x_1+2x_3=0 \\ x_2-3x_3=0 \end{array} \right.

    We can express x_1 and x_2 in terms of x_3 so choose x_3=t.
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  5. #5
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    Re: Finding the N(F) and V(f) the kernel and a values of a matrix, Questions

    I get what you mean by a one zero row now, thanks

    I must ask though if you know the reason for choosing dim (Vf) = 2.

    Are you guys familiar with this terminology?
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  6. #6
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    Re: Finding the N(F) and V(f) the kernel and a values of a matrix, Questions

    Terminology? Even if you are studying Linear Algebra on your own, you should have seen the definition of "dimension of a vector space". Here, as has been pointed out, row reduction shows that the original system of equations, giving the kernel of F, is equivalent to x_1+ 2x_3= 0 and x_2- 3x_3= 0 which are, of course, equivalent to x_1= -2x_3 and x_2= 3x_3. That means that any vector in the solution space (the kernel of the linear transformation) is of the form [tex]<x_1, x_2, x_3>= <-2x_3, 3x_3, x_3>= x_3<-2, 3, 1>. That is, any vector in the kernel is simply <-2, 3, 1> times a constant. <-2, 3, 1> spans the entire space and, of course, a set containing a single vector is "independent" so it is a basis for the kernel- the kernel has dimension 1.

    There is a theorem, that perhaps you have not met yet, that says that if F is a linear transformation from vector space U to vector space V, the sum of the dimensions of the kernel and image is equal to the dimension of U. Here U has dimension 3 and the kernel has dimension 1 so the image has dimension 2. That is, F maps all of R^3 into a 2 dimensional subspace of R^3.

    If you have not had that theorem, you can prove that the image has dimension 2 directly: let <a, b, c> be some vector in the image. Then 2x_1+ 4x_3= a, x_1+ x_2- x_3= b, and -x_1+ 3x_2- 11x_3= c for some x_1, x_2, x_3. We can put that into an augmented matrix and row reduce:
    \begin{bmatrix}2 & 0 & 4 & a \\ 1 & 1 & -1 & b \\ -1 & 3 & -11 & c\end{bmatrix}
    \begin{bmatrix}1 & 1 & -1  & b \\ 0 & 1 & -3 & b- \frac{a}{2}\\ 0 & 0 & 0 & c- 2b+ 2a\end{bmatrix}
    which is solvable if and only if c- 2b+ 2a= 0 or c= 2b- 2a. That is, any such vector can be written <a, b, c>= <a, b, 2b- 2a>= <a, 0, -2a>+ <0, 1, 2b>= a<1, 0, -2>+ b<0, 1, 2>. Those two vectors span the image of F and are independent so are a basis- the image of F has dimension 2.
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  7. #7
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    Re: Finding the N(F) and V(f) the kernel and a values of a matrix, Questions

    Thank you, I am studying linear alg in another language than English, thats why am confused about the terminology
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