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Chapter 5: Problem 3

Show that if \(\overrightarrow{\mathbf{v}}_{1}, \vec{v}_{k}\) are linearly dependent, \(\operatorname{vol}_{k}\left(P\left(\vec{v}_{1}, \ldots, \vec{v}_{k}\right)\right)=0\)

Short Answer

Expert verified
The volume is zero because linearly dependent vectors create a degenerate parallelotope.

Step by step solution

01

Understand the Problem

We need to prove that if the vectors \(\vec{v}_1, \vec{v}_k\) are linearly dependent, the k-dimensional volume (or Lebesgue measure) of the k-parallelotope formed by these vectors is zero.
02

Define Linear Dependence

Vectors \(\vec{v}_1, \vec{v}_k\) are linearly dependent if there exist scalars \(a_1, a_2, \ldots, a_k\), not all zero, such that \(a_1\vec{v}_1 + a_2\vec{v}_2 + \ldots + a_k\vec{v}_k = \vec{0}\). In such a case, one of the vectors can be expressed as a linear combination of the others.
03

Determine Geometry Implication

If the vectors are linearly dependent, then one of the vectors, say \(\vec{v}_k\), can be written as a combination of the others. This means that \(P(\vec{v}_1, \ldots, \vec{v}_k)\) is a degenerate parallelotope, collapsing into a lower-dimensional space.
04

Evaluate Volume of Parallelotope

The volume of a k-dimensional object that is essentially contained within a subspace of lower than k dimensions is zero. Specifically, any degenerate k-dimensional parallelotope formed by linearly dependent vectors has zero volume.
05

Conclude with the Volume Formula

Using the volume formula for a k-dimensional parallelotope \( \operatorname{vol}_k(P(\vec{v}_1, \ldots, \vec{v}_k)) \) involves the determinant of the matrix formed by these vectors. A linearly dependent set implies this matrix is singular (i.e., has zero determinant), hence the volume is zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Spaces
A vector space, in its simplest form, is a collection of vectors, which can be added together and multiplied by scalars. This is what mathematicians call a set along with two operations. The operations are vector addition and scalar multiplication.

Vector spaces have a few important properties:
  • Commutativity: You can add vectors in any order.
  • Associativity: Groups of vectors can be added together in any sequence without changing the outcome.
  • Existence of zero vector: There is always a zero vector that, when added to any vector, does not change the vector.
  • Inverse for every vector: For every vector, there is another that combines with it to form the zero vector.
  • Distributivity: Scalar multiplication is distributive with respect to vector addition and scalar addition.
When solving problems in vector spaces, understanding these properties is key to predicting the behavior of vectors, such as when evaluating linear dependence, which leads us to the interesting case of degenerate parallelotopes.
Volume of Parallelotope
A parallelotope is a geometric object in any dimension that is analogous to a parallelogram in two dimensions and a parallelepiped in three dimensions. When you take vectors from a vector space and use them to form a shape, you get a parallelotope.

The most fascinating property of parallelotopes is their volume. The k-dimensional volume of a parallelotope constructed by k vectors can be understood as a measure of how much space the shape occupies in its n-dimensional space where the vectors reside.
  • If the vectors are linearly independent, the volume of the parallelotope is non-zero.
  • If the vectors are linearly dependent, the object becomes degenerate. A degenerate parallelotope is one that collapses onto a lower-dimensional subspace, thus having no interior volume.
Understanding the concept of the volume of parallelotopes helps in visualizing how linearly dependent vectors affect the geometry of shapes in vector spaces.
Determinants
Determinants are central to many areas of mathematics, especially in the study of vector spaces and matrices. In essence, the determinant is a special number calculated from a square matrix and it has meaningful geometric interpretations when applied to vectors.

Here are key aspects of determinants:
  • Determinants help to determine if a set of vectors is linearly independent.
  • They can be used to Calculate the volume of parallelotopes.
  • A matrix with a zero determinant is called singular, indicating that the vectors used to form this matrix are linearly dependent.
When you take the determinant of a matrix formed by vectors in a vector space, you essentially capture the volume of the parallelotope formed by these vectors.
A zero determinant tells us that the k-dimensional parallelotope is degenerate, collapsed in lower dimensions, confirming zero volume.
Lebesgue Measure
Lebesgue measure is an important concept in mathematical analysis, well-suited for handling the complexities of measuring volumes in higher-dimensional spaces. Unlike simple length, area, or volume calculations, Lebesgue measure gives a real value reflecting the size of a set within a given space and is particularly useful for advanced concepts and irregular shapes.

Key points about Lebesgue measure include:
  • It generalizes the concept of volume to more abstract, complex geometrical shapes.
  • It is useful in spaces that aren鈥檛 easily described by simple geometrical shapes.
  • In the case of a degenerate k-dimensional parallelotope (due to linearly dependent vectors), Lebesgue measure proves the volume is zero.
When used in the context of linear algebra, Lebesgue measure reaffirms conclusions about the zero volume of degenerate shapes, which serves as a bridge between analysis and algebra via vector spaces and determinants.

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Most popular questions from this chapter

(a) Let \(\left(\begin{array}{l}r(t) \\ \theta(t)\end{array}\right)\) be a parametrization of a curve in polar coordinates. Show that the length of the piece of curve between \(t=a\) and \(t=b\) is given by the integral $$ \int_{a}^{b} \sqrt{\left(r^{\prime}(t)\right)^{2}+(r(t))^{2}\left(\theta^{\prime}(t)\right)^{2}} d t $$ (b) What is the length of the spiral given in polar coordinates by $$ \left(\begin{array}{l} r(t) \\ \theta(t) \end{array}\right)=\left(\begin{array}{c} e^{-\alpha t} \\ t \end{array}\right), \alpha>0 $$ between \(t=0\) and \(t=a\) ? What is the limit of this length as \(\alpha \rightarrow 0\) ? (c) Show that the spiral turns infinitely many times around the origin as \(t \rightarrow \infty\). Does the length tend to \(\infty\) as \(t \rightarrow \infty\) ?

(a) Give a parametrization of the surface of the ellipsoid \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{\mathrm{c}^{2}}=1 \quad\) analogous to spherical coordinates. (b) Set up an integral to compute the surface area of the ellipsoid.

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a smooth positive function. Find a parametrization for the surface of equation $$ \frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=(f(z))^{2} $$

Compute the area of the graph of the function \(f\left(\begin{array}{l}x \\\ y\end{array}\right)=\frac{2}{3}\left(x^{3 / 2}+y^{3 / 2}\right)\) above the region \(0 \leq x, y \leq 1\).

Compute \(\int_{S}\left(x^{2}+y^{2}+3 z^{2}\right)\left|d^{2} \mathbf{x}\right|\), where \(S\) is the part of the paraboloid of revolution \(z=x^{2}+y^{2}\) where \(z \leq 9\)
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