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行列式与体积

1. Introduction

Introduction

2. Parallelograms and Paralellepipeds

Parallelograms and Paralellepipeds

3. Determinants and Volumes

Determinants and Volumes

4. Volumes of Regions

Volumes of Regions

5. Which parallelogram is flat?

Each pair of vectors below determines a parallelogram in $\mathbb{R}^2$ . Which pair determines a parallelogram of area $0$ ?

6. What a negative determinant means

A $3\times 3$ matrix $A$ has $\det(A) = -2$ , and $T(x) = Ax$ is the corresponding matrix transformation. Which statement about the image of the unit cube is correct?

7. Area from the columns

A $2\times 2$ matrix $A$ has $\det(A) = 7$ . What is the area of the parallelogram determined by the columns of $A$ ? Enter a number.

8. Area of a triangle

Find the area of the triangle with vertices $(0,0)$ , $(3,1)$ and $(1,4)$ . Enter a number or a fraction, for example 3/2.

9. Area factor of polar coordinates

The polar change of variables is $x = r\cos\theta$ , $y = r\sin\theta$ , with Jacobian matrix

\left(\begin{array}{cc} \partial x/\partial r & \partial x/\partial\theta \\ \partial y/\partial r & \partial y/\partial\theta \end{array}\right).

Enter the determinant of that matrix - the factor by which the change of variables scales area - as an expression in $r$ .

10. Area scaling by a determinant

Let $T(x) = Ax$ be the matrix transformation of a $2\times 2$ matrix $A$ . What does the number $|\det(A)|$ measure?

11. Compute an area scale

Let

A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}.

By what factor does $T(x) = Ax$ scale areas in the plane?

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