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...
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=1The Attempt at a Solution
I have no idea what to do...
Thanks!
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...
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...
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...
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...
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...
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.
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...