Exploration Discussion
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Exploration_AreasandVolume.pdf
EXPLORATION OF MATRIX ALGEBRA AND
DETERMINANTS
Introduction Computer Graphics Areas and Volumes Check Understanding Discuss! References
A���� ��� V������ U���� D����������� �� C������ A���� ��� V������ Determinants of 2 by 2 and 3 by 3 matrices have a geometric interpretation in terms of the areas and volumes of the two- or three-dimensional figures being transformed. In particular, the determinant is a measure of areas and volumes, as seen in the following theorem.
Theorem 2.2.1. Given a 2 by 2 matrix A whose column vectors represent two adjacent sides of a parallelogram, the area of the parallelogram.
Similarly, given a 3 by 3 matrix A whose column vectors represent three edges of a parallelepiped emanating from the same vertex, the volume of the parallelepiped.
The following video gives a short geometric proof for the 2 by 2 case. The 3 by 3 case can be proven similarly.
Video 2.2.1: Proof of Theorem 2.2.1
|det A| =
|det A| =
Listen
Video by Dr. Lisa Korf: Proof of Theorem 2.2.1 [© CCCOnline]
Video 2.2.1 Transcript: Proof of Theorem 2.2.1
E������ ₂.₂.₁: C���������� ��� ���� �� � ������������� ����� ��� ����������� In the next video example, we'll apply Theorem 2.2.1 to calculate the area of the parallelogram with vertices given by the points (2, –2), (–3, 0), (1, 4) and (–4, 6).
Video 2.2.2: Using the Determinant to Compute the Area of a Parallelogram
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Video by Dr. Lisa Korf: Using the Determinant to Computer the Area of a Parallelogram [© CCCOnline]
Video 2.2.2 Transcript: Using the Determinant to Compute the Area of a Parallelogram
T�� E����� �� � L����� T������������� �� A���� ��� V������ Determinants can also be used to calculate how the area or volume of a figure changes when a linear transformation is applied to it. Theorem 2.2.2. For a linear transformation T determined by a 2 by 2 matrix A , if S is a figure in R3 with a finite area such as a disk, then:
For a linear transformation T determined by a 3 by 3 matrix A, if S is a figure in R3 with a finite volume such as a ball, then:
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Area of T (S) = |det A| ⋅Area of S.
Volume of T (S) = |det A| ⋅ Volume of S.
E������ ₂.₂.₂: U���� ��� ����������� �� ������� ��� ���� ������� �� �� ������� The next example illustrates how Theorem 2.2.2 can be used to compute the area bounded by an ellipse.
Video 2.2.3: Computing the Area Bounded by an Ellipse
Video by Dr. Lisa Korf: Computing the Area Bounded by an Ellipse [© CCCOnline]
Video 2.2.3 Transcript: Computing the Area Bounded by an Ellipse
This same technique could be extended to compute the volume bounded by an ellipsoid. Check your understanding next in the Check Understanding tab.
Introduction Computer Graphics Areas and Volumes Check Understanding Discuss! References
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