Determinants as Area or Volume

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Homework Help Overview

The problem involves finding the volume of a transformed parallelepiped in three-dimensional space, defined by three vectors. The transformation is represented by a matrix, and the relationship between the volume of the original solid and the transformed solid is expressed through the determinant of the matrix.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the definitions of the vectors and their roles in defining both the solid and the transformation. There are questions about the calculations of the determinant and its implications for the volume of the transformed solid.

Discussion Status

Some participants are verifying the problem statement and the calculations of the determinant. There is a divergence in the calculated values of the determinant, leading to further questioning of the assumptions and interpretations of the problem.

Contextual Notes

Participants note potential discrepancies in the calculated determinant values and the expected volume, suggesting a need for clarification on the problem setup and definitions.

Lonely Lemon
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Homework Statement



If S is a parallelepiped determined by v1=(1, 1, 0) and v2= (3, 2, 1) and v3=(6, 1, 2) and T: R3--> R3 by T(x)=Ax, find the volume of T(S)

Homework Equations



{volume of T(S)}=|det A|.{volume of S}

The Attempt at a Solution



A is [v1 v2 v3] and the |A| = 9 by my calculations. I thought this was the volume, but the answer to the questions is given as 24. Please help!
 
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So the vectors define both the solid S and the mapping A? Just want to make sure you've described the problem as assigned.
 
I assume so, that's the problem posed word for word. T: R3-->R3 is the linear transformation determined by a 3x3 matrix A, and S is the a parallelepiped in R3, so the vectors define both?
 
If that's true, then shouldn't the volume of T(S) be a perfect square? By the way, I get det A=3, so you might want to recheck your calculations.
 

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