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    Define a linear transformation. T: P_2 --> R^2 by

    T(p) = \begin{bmatrix} p(0) \\p(0) \end{bmatrix}.

    Find polynomials p_1 and p_2 in P_2 that span the kernel of T, and describe the range of T.

    My answer:

    This is the definition of a linear transformation:

    T(p+q) = \begin{bmatrix} (p+q)(0) \\(p+q)(0) \end{bmatrix}

    = \begin{bmatrix} p(0) \\p(0) \end{bmatrix} + \begin{bmatrix} q(0) \\q(0) \end{bmatrix}

    = T(p) + T(q)

    T(cp) = \begin{bmatrix} (cp)(0) \\(cp)(0) \end{bmatrix}

    = c\begin{bmatrix} p(0) \\p(0) \end{bmatrix}= cT(p)

    These are the polynomials p_1 and p_2 in P_2:

    T(p) = \begin{bmatrix} p(0) \\p(0) \end{bmatrix} = \begin{bmatrix} 0 \\0 \end{bmatrix}

    So p(0) = 0 and p(0) = 0...

    I chose any two polynomials that met these requirments...so...

    p_1 = x^2
    p_2 = 2x^2

    Range: R^2
    Is my answer correct?

    Thank you in advance.
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    (Original post by Artus)
    I chose any two polynomials that met these requirments...so...

    p_1 = x^2
    p_2 = 2x^2

    The range of T is R^2, because there are two rows in T(p)...

    Is my answer correct?

    Thank you in advance.
    If P_2 is the set of polynomials of degree at most 2, then

    consider p=x. Is this in the kernal of T? And if so what's its representation in terms of the span you've proposed?
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    (Original post by ghostwalker)
    If P_2 is the set of polynomials of degree at most 2, then

    consider p=x. Is this in the kernal of T? And if so what's its representation in terms of the span you've proposed?
    Yes it is in the kernal of T....what span do you mean? I didn't propose any span...
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    (Original post by Artus)
    Yes it is in the kernal of T....what span do you mean? I didn't propose any span...
    In that case, I've no idea what you're doing with that P_1 and p_2.
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    (Original post by ghostwalker)
    In that case, I've no idea what you're doing with that P_1 and p_2.
    Oh...ok I get it now...yes I did propose a span....I'm sorry, I just noticed it ...

    Yes, p = x will be in the span, because p(0) = 0 and p(0)=0...and that's in the span so...yes...

    p = 1x + 0x ... Is that right?
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    (Original post by Artus)

    p = 1x + 0x ... Is that right?
    Right for what? I've no idea what you're trying to say with that.

    Since "x" is in the kernal, it must be representably as a linear sum of the vectors that span the kernal, i.e. the p_1 and p_2 that you've proposed. Is it?
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    (Original post by ghostwalker)
    Right for what? I've no idea what you're trying to say with that.

    Since "x" is in the kernal, it must be representably as a linear sum of the vectors that span the kernal, i.e. the p_1 and p_2 that you've proposed. Is it?
    I'm sorry, but I don't know what you mean....the question asks me to "Find polynomials p1 and p2 in P_2 that span the kernel of T"...since p(0)=0 for both entries...it means that any equation like p(x) = ax^2 will span the kernel...so I said p1 = x^2 and p2=2x^2... I could not understand what you want me to do with p = x ... can you clarify that please?
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    (Original post by Artus)
    I'm sorry, but I don't know what you mean....the question asks me to "Find polynomials p1 and p2 in P_2 that span the kernel of T"...since p(0)=0 for both entries...it means that any equation like p(x) = ax^2 will span the kernel...so I said p1 = x^2 and p2=2x^2... I could not understand what you want me to do with p = x ... can you clarify that please?
    You really need to go back to your text book and see what these terms mean and look at some elemetary examples.

    If a set (A) of vectors spans a space (or subspace, which is what the kernal is), then every vector in that space (or subspace) can be written as a linear sum of the elements of A. That's what "spans" means.

    E.g. in R^2, which can be represented by ordered pairs whose elements are in R, then every element can be written in terms of the spanning set {(1,0), (0,1)} for example.

    If you had the element (3,5), then it can be written as 3(1,0) + 5(0,1) etc.
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    (Original post by ghostwalker)
    You really need to go back to your text book and see what these terms mean and look at some elemetary examples.

    If a set (A) of vectors spans a space (or subspace, which is what the kernal is), then every vector in that space (or subspace) can be written as a linear sum of the elements of A. That's what "spans" means.

    E.g. in R^2, which can be represented by ordered pairs whose elements are in R, then every element can be written in terms of the spanning set {(1,0), (0,1)} for example.

    If you had the element (3,5), then it can be written as 3(1,0) + 5(0,1) etc.
    Actually I did read the textbook...but its the first time I'm studying this, so its not clear in my head...that's why I'm asking...is my answer right or wrong?
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    (Original post by Artus)
    Actually I did read the textbook...but its the first time I'm studying this, so its not clear in my head...that's why I'm asking...is my answer right or wrong?
    Well you posted a thread over a week ago on spans and virtually had it correct then, and now seem to have forgotten what it means.

    Your answer is wrong by the way.

    You don't seem to have an understanding of what you're doing, and whilst this forum is here to help people, it's not a teaching forum, though some people are very generous in that department.

    And referring back to your first post, the range of T is not R^2.

    Seriously, you need to go back and get these concepts under your belt; just reading them won't help your understanding unless you look at examples in detail, making sure you understand them and do basic exercises with them.

    The one you've posted isn't a difficult exercise.
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    (Original post by ghostwalker)
    Well you posted a thread over a week ago on spans and virtually had it correct then, and now seem to have forgotten what it means.

    Your answer is wrong by the way.

    You don't seem to have an understanding of what you're doing, and whilst this forum is here to help people, it's not a teaching forum, though some people are very generous in that department.

    And referring back to your first post, the range of T is not R^2.

    Seriously, you need to go back and get these concepts under your belt; just reading them won't help your understanding unless you look at examples in detail, making sure you understand them and do basic exercises with them.

    The one you've posted isn't a difficult exercise.
    I wasn't asking anybody to teach me; I was just asking if my answer was right or wrong...thanks for answering by the way...I thought it was right, because there was another question that was 90% similar to mine, and they solved it the same way...and by the way, the question I asked a week ago was different. If you had asked me that question, I would have known how to do it...by the way I'm not just reading, I'm studying. You should not jump to conclusions about what I'm doing when you don't know what I'm doing. If you feel that you have the time to answer my questions then you can, otherwise you don't need to. Anyways, thanks for answering.
 
 
 
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