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Chapter 2: Problem 14

Describe the span of the given vectors (a) geometrically and (b) algebraically. $$\left[\begin{array}{l} 0 \\ 0 \end{array}\right],\left[\begin{array}{l} 3 \\ 4 \end{array}\right]$$

Short Answer

Expert verified
The span is a line through the origin along \( \begin{bmatrix} 3 \\ 4 \end{bmatrix} \).

Step by step solution

01

Introduce the Vectors

We have two vectors: \( \mathbf{v_1} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \) and \( \mathbf{v_2} = \begin{bmatrix} 3 \ 4 \end{bmatrix} \). These vectors are in a 2-dimensional space.
02

Understanding Span Geometrically

Geometrically, the span of a set of vectors is the set of all points that can be reached by a linear combination of the vectors. The zero vector alone, \( \mathbf{v_1} \), does not contribute to the span direction as it represents a point at the origin. The vector \( \mathbf{v_2} \) determines a line through the origin in the direction of \( \begin{bmatrix} 3 \ 4 \end{bmatrix} \). Thus, the span forms a line.
03

Understanding Span Algebraically

Algebraically, the span of \( \mathbf{v_1} \) and \( \mathbf{v_2} \) is the set of all vectors that can be written as a linear combination of \( \mathbf{v_1} \) and \( \mathbf{v_2} \), i.e., \( c_1 \cdot \mathbf{v_1} + c_2 \cdot \mathbf{v_2} \) where \( c_1 \) and \( c_2 \) are scalars. Since \( \mathbf{v_1} \) contributes nothing (it is zero), the span reduces to vectors of the form \( c_2 \cdot \begin{bmatrix} 3 \ 4 \end{bmatrix} \), which represents a line through the origin along \( \mathbf{v_2} \).
04

Conclusion

The geometric interpretation involves a line through the origin in the direction given by the vector \( \begin{bmatrix} 3 \ 4 \end{bmatrix} \). Algebraically, it consists of all vectors that are scalar multiples of \( \begin{bmatrix} 3 \ 4 \end{bmatrix} \), confirming that the span is a line.

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Key Concepts

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

Geometric Interpretation
Vectors in geometry are depicted as arrows in a coordinate space, originating from the origin or another point. In this context, the **geometric interpretation** of the span of vectors focuses on what regions or lines these vectors create in space. For the vectors \( \mathbf{v_1} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \) and \( \mathbf{v_2} = \begin{bmatrix} 3 \ 4 \end{bmatrix} \):
  • \( \mathbf{v_1} \) is the zero vector. It represents a point at the origin (0,0) and does not define a direction or span any line or plane.
  • \( \mathbf{v_2} \) is a vector that points towards the coordinates (3,4). This creates a straight line when extended in both directions through the origin.
This line represents the geometric span of the vectors, as all reachable points through linear combinations of these vectors will lie on this line.
Algebraic Representation
The **algebraic representation** of the span of vectors describes all the possible vector combinations that can be formed using a set of given vectors. To understand the span through algebraic terms, consider any linear combination of vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \):
  • Any vector in the span can be expressed as \( c_1 \cdot \mathbf{v_1} + c_2 \cdot \mathbf{v_2} \), with \( c_1 \) and \( c_2 \) being scalar values.
  • Specifically, because \( \mathbf{v_1} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \), the zero vector contributes nothing to the directionality or extent of the span.
  • This simplifies the algebraic expression to \( c_2 \cdot \begin{bmatrix} 3 \ 4 \end{bmatrix} \).
Thus, all vectors in this span are scalar multiples of \( \begin{bmatrix} 3 \ 4 \end{bmatrix} \), forming a line through the origin.
Linear Combinations
**Linear combinations** involve combining vectors together using scalar multiplication and addition. The concept is central to understanding spans and other vector space elements. The process of forming linear combinations is easy:
  • Each given vector is multiplied by a scalar (a real number), which stretches or shrinks the vector.
  • The results from this multiplication are then added together to form a new vector.
In our example with vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \):
  • The expression \( c_1 \cdot \begin{bmatrix} 0 \ 0 \end{bmatrix} + c_2 \cdot \begin{bmatrix} 3 \ 4 \end{bmatrix} \) uses both operations to combine the vectors.
  • Here, since \( c_1 \cdot \begin{bmatrix} 0 \ 0 \end{bmatrix} \) remains zero, the overall equation simplifies based on the scaling of \( \mathbf{v_2} \) by scalar \( c_2 \).
In essence, linear combinations depict how different vectors can be added together to span a line, plane, or higher-dimensional space.

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Most popular questions from this chapter

Let \(\left\\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{k}\right\\}\) be a linearly independent set of vectors in \(\mathbb{R}^{n}\), andlet v be a vector in \(\mathbb{R}^{n}\). Suppose that \(\mathbf{v}=c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}+\cdots+c_{k} \mathbf{v}_{k}\) with \(c_{1} \neq 0 .\) Prove that \(\left\\{\mathbf{v}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\\}\) is linearly independent

(a) If vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are linearly independent, will \(\mathbf{u}+\mathbf{v}, \mathbf{v}+\mathbf{w},\) and \(\mathbf{u}+\mathbf{w}\) also be linearly independent? Justify your answer. (b) If vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are linearly independent, will \(\mathbf{u}-\mathbf{v}, \mathbf{v}-\mathbf{w},\) and \(\mathbf{u}-\mathbf{w}\) also be linearly independent? Justify your answer.

Let \(\mathbf{p}=\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right], \mathbf{q}=\left[\begin{array}{r}0 \\ 1 \\ -1\end{array}\right], \mathbf{u}=\left[\begin{array}{r}2 \\ -3 \\ 1\end{array}\right],\) and \(\mathbf{v}=\left[\begin{array}{r}0 \\ 6 \\ -1\end{array}\right]\) Show that the lines \(\mathbf{x}=\mathbf{p}+\) su and \(\mathbf{x}=\mathbf{q}+t \mathbf{v}\) are skew lines. Find vector equations of a pair of parallel planes, one containing each line.

Solve the given system of equations using either Gaussian or Gauss-Jordan elimination. \(2 w+3 x-y+4 z=1\) \(3 w-x+z=1\) \(3 w-4 x+y-z=2\)

In Gotham City, the departments of Administration (A), Health (H), and Transportation (T) are interdependent. For every dollar's worth of services they produce, each department uses a certain amount of the services produced by the other departments and itself, as shown in Table \(2.10 .\) Suppose that, during the year, other city departments require \(\$ 1\) million in Administrative services, \(\$ 1.2\) million in Health services, and \(\$ 0.8\) million in Transportation services. What does the annual dollar value of the services produced by each department need to be in order to meet the demands?
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