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Magazine Chapter 1: Problem 24
In Exercises 23 and \(24,\) mark each statement True or False. Justify each answer. a. If \(\mathbf{x}\) is a nontrivial solution of \(A \mathbf{x}=\mathbf{0},\) then every entry in \(\mathbf{x}\) is nonzero. b. The equation \(\mathbf{x}=x_{2} \mathbf{u}+x_{3} \mathbf{v}\) , with \(x_{2}\) and \(x_{3}\) free (and neither \(\mathbf{u}\) nor \(\mathbf{v}\) a multiple of the other), describes a plane through the origin. c. The equation \(A \mathbf{x}=\mathbf{b}\) is homogeneous if the zero vector is a solution. d. The effect of adding \(\mathbf{p}\) to a vector is to move the vector in a direction parallel to \(\mathbf{p}\) .
Short Answer
Expert verified
a. False, b. True, c. True, d. True.
Step by step solution
01
Understanding Statement a
Consider the equation \(A \mathbf{x} = \mathbf{0}\). A nontrivial solution exists if there are free variables in the solution set of \(\mathbf{x}\). In this case, \(\mathbf{x}\) can have zero entries as long as it is not the zero vector itself. Therefore, not all entries of \(\mathbf{x}\) must be nonzero.
02
Assessing Statement a
Statement a is False because it is possible for \(\mathbf{x}\) to have some zero entries in a nontrivial solution, while not being the zero vector itself. Nontrivial implies simply not the zero vector, but does not restrict individual components.
03
Understanding Statement b
The equation \(\mathbf{x} = x_{2} \mathbf{u} + x_{3} \mathbf{v}\), where \(x_{2}\) and \(x_{3}\) are free variables, represents a linear combination of \(\mathbf{u}\) and \(\mathbf{v}\). Since \(\mathbf{u}\) and \(\mathbf{v}\) are independent vectors and linear combinations in two free variables describe a plane through the origin, the statement is geometrically feasible.
04
Assessing Statement b
Statement b is True because the linear combination of two independent vectors with free scalar multiples represents a plane through the origin. Neither \(\mathbf{u}\) nor \(\mathbf{v}\) being a multiple of the other ensures their independence.
05
Understanding Statement c
A system of equations \(A \mathbf{x} = \mathbf{b}\) is homogeneous if it can be written in the form \(A \mathbf{x} = \mathbf{0}\). If the zero vector is a solution, \(\mathbf{b}\) must be the zero vector, thus indicating a homogeneous system.
06
Assessing Statement c
Statement c is True because if the zero vector is a solution, then \(A \mathbf{x} = \mathbf{0}\), and hence the system must be homogeneous.
07
Understanding Statement d
Adding a vector \(\mathbf{p}\) to another vector results in a translation parallel to \(\mathbf{p}\). The direction and distance of the translation correspond to \(\mathbf{p}\), embodying the concept of vector addition.
08
Assessing Statement d
Statement d is True because adding \(\mathbf{p}\) to a vector translates the vector along a line parallel to \(\mathbf{p}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Nontrivial Solution
In linear algebra, a **nontrivial solution** refers to a solution to a homogeneous equation that is not the zero vector. When you have a system of equations, like \[ A \mathbf{x} = \mathbf{0} \]where \(A\) is a matrix and \(\mathbf{x}\) is a vector, the trivial solution is simply when \(\mathbf{x}\) is the zero vector. However, the interesting cases arise when there are nontrivial solutions, meaning \(\mathbf{x}\) is not the zero vector.
- Nontrivial solutions occur when there are free variables in the system, which allow for multiple possible solutions.
- It's possible for some components of \(\mathbf{x}\) to be zero, as long as the entire vector isn't the zero vector.
- Such solutions indicate a degree of freedom or redundancy in the given matrix system, often seen in terms of linear combinations of vectors.
Homogeneous Equations
**Homogeneous equations** are systems where every linear equation in the system equals zero. Mathematically, they are represented as:\[ A \mathbf{x} = \mathbf{0} \]Since the zero vector is always a solution, homogeneous systems are guaranteed to have at least the trivial solution.
- The existence of nontrivial solutions depends on the rank of the matrix; specifically, if the number of columns exceeds the rank, free variables are present.
- Homogeneous equations are significant because they fully describe the null space of the matrix \(A\).
- The solution space of homogeneous systems represents all possible linear combinations that result in the zero vector, effectively describing dependencies among vectors.
Vector Addition
**Vector addition** in linear algebra involves combining two or more vectors to produce another vector. This process is crucial for understanding vector operations and their applications in space.
- When you add vector \(\mathbf{p}\) to another vector \(\mathbf{q}\), the resulting vector is a translation of \(\mathbf{q}\) along the direction of \(\mathbf{p}\).
- The operation is performed component-wise: for vectors \(\mathbf{a} = (a_1, a_2)\) and \(\mathbf{b} = (b_1, b_2)\), the sum is \(\mathbf{a} + \mathbf{b} = (a_1 + b_1, a_2 + b_2)\).
- Vector addition is both commutative and associative, properties that facilitate the manipulation and combination of vectors in systems.
Linear Combination
A **linear combination** in linear algebra is any expression constructed from a set of terms by multiplying each term by a constant and then adding the results. Specifically, for vectors \(\mathbf{u}\), \(\mathbf{v}\), and scalars \(x_2\), \(x_3\):\[ \mathbf{x} = x_2 \mathbf{u} + x_3 \mathbf{v} \]Here, \(x_2\) and \(x_3\) are coefficients or scalar multipliers.
- If \(\mathbf{u}\) and \(\mathbf{v}\) are independent, any linear combination of these vectors covers a plane through the origin.
- Linear combinations are fundamental in forming vector spaces and understanding span, which represents all possible vectors achievable through such combinations.
- In practical terms, linear combinations allow for the representation of complex systems in terms of a smaller, manageable set of vectors.
Independent Vectors
**Independent vectors** in the context of linear algebra are vectors that cannot be expressed as a linear combination of one another. If \(\mathbf{u}\) and \(\mathbf{v}\) are independent, neither is a scalar multiple of the other.
- Vectors are independent if the only solution to the equation \[ x_2 \mathbf{u} + x_3 \mathbf{v} = \mathbf{0} \] is when \(x_2 = x_3 = 0\).
- Independence implies that no vector in the set can be formed by scaling and adding other vectors from the same set, showcasing their non-redundancy.
- This concept is pivotal in defining bases; a set of independent vectors can span a vector space, providing a foundation for all possible vectors in that space.
