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    Abstract Algebra - ideals and generators

    Homework Statement a.) Let a=3-8i and b=2+3i. Find x,y ϵ Z[i] such that ax+by=1. b.) Show explicitly that the ideal I=(85,1+13i) \subseteq Z[i] is principle by exhibiting a generator. Homework Equations Given ideal: I=(85,1+13i) \subseteq Z[i] a=3-8i b=2+3i Honestly, I am beyond lost...
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    Tangent plane question

    I'm completely lost on this question, and it's due tomorrow morning. Help? Homework Statement What point on y=x2+z2 is the tangent plane parallel to the plane x+2y+3z=1? Homework Equations y=x2+z2 x+2y+3z=1 The Attempt at a Solution I have no idea what to do... Thanks!
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    Linear independence?

    Ahhh. And by "components" you mean rows?
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    Linear independence?

    What do you mean by "3 components"?
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    Linear independence?

    How do you know when a matrix (or equivocally a system of equations) is linearly independent? How do you know that it's linearly dependent? For example, given this matrix, [ 1 1 2 1] [-2 1 4 0] [ 0 3 2 2] How do we know if this matrix is linearly independent or dependent...
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    Free variables in linear algebra?

    Ohhhh OK. So, to clarify, in this matrix we have 5 columns (but only four of which have constants in them), so we have four variables. Then we have three rows (each row equals an equation). So we have 4 variables minus 3 rows which is equal to 1 free variable. We don't want to choose any...
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    Free variables in linear algebra?

    My questions is short and to the point: What exactly is a free variable (in a matrix, for example). How do you know if a variable is free? Here's a matrix that (apparently) has a free variable: [1 4 -3 0 0] [-2 -7 5 1 0 ] [-4 -5 7 5 0 ] Row reducing the matrix we end up with: [1 4 -3 0 0...
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    Question about solving augmented matrices and row operations

    Oh OK. Thanks so much for your help!! :)
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    Question about solving augmented matrices and row operations

    Then what would happen if we had two non-zero coefficients in the same column? Like, [2 1 | 1] [3 0 | 2] Would that be 2x1+1x2=1 and 3x1+0x2=2??
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    Question about solving augmented matrices and row operations

    OK, so it appears that each coefficient on the left side of the matrix gets it's own variable (x1, x2, x3, etc.) So if we had 6 coefficients in the matrix, we'd have variables x1->6? For example, given the matrix: [2 3 4 | 1] [1 0 7 | 2] [0 0 5 | 3] we'd say that 2x1+3x2+4x3=1...
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    Question about solving augmented matrices and row operations

    Ohhhh OK. So it's sort of like solving a simple system of linear equations by the elimination method, right? So I suppose my next question is how do we know when to stop simplifying the matrices? I see in the final matrix that the third row is completely eliminated (it's all zeros). Also...
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    Question about solving augmented matrices and row operations

    Well, I understand what they mean. I don't understand how we know that in order to get R2 in the second matrix we have to add R2+4R3 from the first matrix. How do we know what to add, subtract, multiply, or divide? How do we know that R2=R2+4R3. That's what I'm confused about.
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    Question about solving augmented matrices and row operations

    So I just started my Linear Algebra course yesterday. I am confused on one aspect. When asked to solve an augmented matrix, the teacher would employ row operations. I understand how the row operations lead from one matrix to the next, but what I don't understand is how we formulate which row...
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    Calculus III or linear Algebra? Which one should I take first?

    OK, thanks! I think I'm leaning more toward LA at this point. :)
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    Calculus III or Linear Algebra? Which one should I take first?

    So you're saying that I should take LA first?
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    Calculus III or Linear Algebra? Which one should I take first?

    Fragment, I think I'll start off with Diff. EQs. Not sure what I want to take after that...
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    Calculus III or Linear Algebra? Which one should I take first?

    Firstly, as you may see, I'm a newbie to this forum so I apologize if I posted this question in the wrong place or violated any rules within this post. :) I have no idea as to which to take first. The course description for Linear Algebra says that "Calculus III is recommended but no...
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    Calculus III or linear Algebra? Which one should I take first?

    Firstly, as you may see, I'm a newbie to this forum so I apologize if I posted this question in the wrong place or violated any rules within this post. :) I have no idea as to which to take first. The course description for Linear Algebra says that "Calculus III is recommended but no required"...
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