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Chapter 4: Problem 18

\- Let \(S\) be a basis for an \(n\) -dimensional vector space \(V\). Show that if \(v_{1}, v_{2}, \ldots, v_{r}\) form a linearly independent set of vectors in \(V\), then the coordinate vectors \(\left(v_{1}\right) s\), \(\left(v_{2}\right) s\), \(\dots,\left(v_{r}\right)_{S}\) form a linearly independent set in \(R^{n}\), and conversely.

Short Answer

Expert verified
Coordinate vectors retain the linear independence of original vectors and vice versa.

Step by step solution

01

Understanding the Vector Space Basis

A basis \( S \) for an \( n \)-dimensional vector space \( V \) is a set of \( n \) vectors such that every vector in \( V \) can be uniquely represented as a linear combination of these basis vectors. When \( v_1, v_2, \ldots, v_r \) are linearly independent vectors in \( V \), none of these vectors can be expressed as a linear combination of the others.
02

Establish the Linear Independence in \( V \)

Given \( v_1, v_2, \ldots, v_r \) are linearly independent, this means that if we have the linear combination \( a_1v_1 + a_2v_2 + \cdots + a_rv_r = 0 \), then \( a_1 = a_2 = \cdots = a_r = 0 \) is the only solution. This is the definition of linear independence in vector space \( V \).
03

Writing as Coordinate Vectors

Each vector \( v_i \) can be expressed as a coordinate vector \( (v_i)_S \) relative to the basis \( S \). This coordinate vector is a unique \( n \)-tuple in \( \mathbb{R}^n \), which associates with the coefficients of the basis vectors that sum to form \( v_i \).
04

Linear Independence of Coordinate Vectors

To show linear independence of \( (v_1)_S, (v_2)_S, \ldots, (v_r)_S \) in \( \mathbb{R}^n \), consider a linear combination \( b_1(v_1)_S + b_2(v_2)_S + \cdots + b_r(v_r)_S = 0 \). If this combination sums to the zero vector, it corresponds to \( b_1v_1 + b_2v_2 + \cdots + b_rv_r = 0 \) in \( V \). Since \( v_1, v_2, \ldots, v_r \) are linearly independent, it follows that \( b_1 = b_2 = \cdots = b_r = 0 \).
05

Conversely, Coordinate Vector Independence Implying Vector Independence

If \( (v_1)_S, (v_2)_S, \ldots, (v_r)_S \) are linearly independent in \( \mathbb{R}^n \), for any linear combination \( b_1(v_1)_S + b_2(v_2)_S + \cdots + b_r(v_r)_S = 0 \), the only solution is \( b_1 = b_2 = \cdots = b_r = 0 \). This implies the original vectors \( v_1, v_2, \ldots, v_r \) in \( V \) are also linearly independent, as shown by the unique solution for coefficients in their linear combinations.

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Key Concepts

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

Vector Space
A vector space is a fundamental concept in linear algebra where vectors live. It is a collection of objects called vectors, which can be added together and multiplied by scalars (numbers). This collection follows specific rules, like having an additive identity (usually the zero vector) and satisfying the associative and commutative properties of addition.
- **Scalars** are real numbers or complex numbers used to scale vectors.- **Vector Addition** and **scalar multiplication** must satisfy certain axioms or rules. When we talk about an \( n \)-dimensional vector space, it means that exactly \( n \) vectors are needed to describe every vector in the space when used as a basis.
Basis
Understanding a basis is essential for mastering vector spaces. A basis of a vector space \( V \) is a set of vectors in \( V \) that:
  • Are linearly independent, meaning no vector in the set can be expressed as a combination of the others.
  • Span the vector space, meaning any vector in the space can be expressed as a combination of these basis vectors.
In an \( n \)-dimensional vector space, the basis must have exactly \( n \) vectors. This means that:- Any vector in the space is uniquely represented by a linear combination of the basis vectors.- The dimension of the vector space is equal to the number of vectors in the basis.
Coordinate Vectors
Coordinate vectors express a vector from a vector space \( V \) in terms of a chosen basis \( S \). For a vector \( v \), its coordinate vector \( (v)_S \) consists of the coefficients needed to express \( v \) as a linear combination of the basis vectors.
- The coordinate vector is an ordered \( n \)-tuple in \( \mathbb{R}^n \) (if the vector space is \( n \)-dimensional).- This transformation from a vector to its coordinate vector is reversible.Coordinate vectors allow us to work with abstract vector spaces by representing them in the more familiar \( \mathbb{R}^n \), making calculations and theoretical work simpler.
Linear Combination
A linear combination involves creating a new vector by adding together scalar multiples of other vectors. If you have vectors \( v_1, v_2, \ldots, v_r \), then any vector that can be written in the form \( a_1v_1 + a_2v_2 + \cdots + a_rv_r \) is a linear combination of the vectors.
  • **Scalars** \( a_1, a_2, \ldots, a_r \) dictate how each vector contributes to the combination.
  • Linear combinations are used to define concepts like linear independence and span.
Matrix and vector equations often rely on linear combinations to express more complicated systems. Recognizing linear combinations helps in simplifying and solving such systems efficiently.

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Most popular questions from this chapter

(a) If \(A\) is a \(3 \times 5\) matrix, then the rank of \(A\) is at most ____________ . Why? (b) If \(A\) is a \(3 \times 5\) matrix, then the nullity of \(A\) is at most ____________ . Why? (c) If \(A\) is a \(3 \times 5\) matrix, then the rank of \(A^{T}\) is at most ____________ . Why? (d) If \(A\) is a \(3 \times 5\) matrix, then the nullity of \(A^{T}\) is at most ____________ . Why?

Determine whether \(T_{1} \circ T_{2}=T_{2} \circ T_{1}\) (a) \(T_{1}: R^{2} \rightarrow R^{2}\) is the orthogonal projection on the \(x\) -axis, and \(T_{2}: R^{2} \rightarrow R^{2}\) is the orthogonal projection on the \(y\) -axis. (b) \(T_{1}: R^{2} \rightarrow R^{2}\) is the rotation through an angle \(\theta_{1}\), and \(T_{2}: R^{2} \rightarrow R^{2}\) is the rotation through an angle \(\theta_{2}\) (c) \(T_{1}: R^{2} \rightarrow R^{2}\) is the orthogonal projection on the \(x\) -axis, and \(T_{2}: R^{2} \rightarrow R^{2}\) is the rotation through an angle $$\boldsymbol{\theta}$$.

In a laboratory experiment, a mouse can choose one of two food types each day, type I or type II. Records show that if the mouse chooses type I on a given day, then there is a \(75 \%\) chance that it will choose type I the next day, and if it chooses type II on one day, then there is a \(50 \%\) chance that it will choose type II the next day. (a) Find a transition matrix for this phenomenon. (b) If the mouse chooses type I today, what is the probability that it will choose type I two days from now? (c) If the mouse chooses type II today, what is the probability that it will choose type II three days from now? (d) If there is a \(10 \%\) chance that the mouse will choose type I today, what is the probability that it will choose type I tomorrow?

To write the coordinate vector for a vector, it is necessary to specify an order for the vectors in the basis. If \(P\) is the transition matrix from a basis \(B^{\prime}\) to a basis \(B\), what is the effect on \(P\) if we reverse the order of vectors in \(B\) from \(v_{1}, \ldots, v_{n}\) to \(v_{n}, \ldots, v_{1}\) ? What is the effect on \(P\) if we reverse the order of vectors in both \(B^{\prime}\) and \(B ?\)

Consider a Markov process with transition matrix State 1 State 2 State 1 State 2 \(\left[\begin{array}{ll}0 & \frac{1}{7} \\ 1 & \frac{6}{7}\end{array}\right]\) (a) What does the entry \(\frac{6}{7}\) represent? (b) What does the entry 0 represent? (c) If the system is in state 1 initially, what is the probability that it will be in state 1 at the next observation? (d) If the system has a \(50 \%\) chance of being in state 1 initially, what is the probability that it will be in state 2 at the next observation?
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