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Chapter 1: Problem 22

Mark each statement True or False. Justify each answer. a. Every matrix transformation is a linear transformation. b. The codomain of the transformation \(\mathbf{x} \mapsto A \mathbf{x}\) is the set of all linear combinations of the columns of \(A.\) c. If \(T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) is a linear transformation and if \(\mathbf{c}\) is in \(\mathbb{R}^{m},\) then a uniqueness question is "Is \(\mathbf{c}\) in the range of \(T ?\) d. A linear transformation preserves the operations of vector addition and scalar multiplication. e. The superposition principle is a physical description of a linear transformation.

Short Answer

Expert verified
a) True, b) False, c) False, d) True, e) True.

Step by step solution

01

Analyze Statement A

The statement is 'Every matrix transformation is a linear transformation.' A matrix transformation from \( \mathbb{R}^n \) to \( \mathbb{R}^m \) is defined as \( T(\mathbf{x}) = A\mathbf{x} \), where \( A \) is an \( m \times n \) matrix. This operation is linear because it satisfies both linearity properties: \( T(\mathbf{u} + \mathbf{v}) = A(\mathbf{u} + \mathbf{v}) = A\mathbf{u} + A\mathbf{v} = T(\mathbf{u}) + T(\mathbf{v}) \) and \( T(c\mathbf{u}) = A(c\mathbf{u}) = cA\mathbf{u} = cT(\mathbf{u}) \) for all vectors \( \mathbf{u}, \mathbf{v} \) and scalar \( c \). Thus, this statement is **True**.
02

Analyze Statement B

The statement is 'The codomain of the transformation \( \mathbf{x} \mapsto A \mathbf{x} \) is the set of all linear combinations of the columns of \( A \).' The codomain of a transformation is the space into which the transformation maps vectors, which in this case is \( \mathbb{R}^m \), not just the set of linear combinations of the columns of \( A \). The set of all linear combinations of the columns of \( A \) is the range (or the image) of the transformation. Therefore, this statement is **False**.
03

Analyze Statement C

The statement is 'If \( T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} \) is a linear transformation and if \( \mathbf{c} \) is in \( \mathbb{R}^{m}, \) then a uniqueness question is "Is \( \mathbf{c} \) in the range of \( T \)?' Uniqueness questions typically ask if a particular output is produced by only one input. Determining whether \( \mathbf{c} \) is in the range of \( T \) is an existence question, not a uniqueness question. Therefore, this statement is **False**.
04

Analyze Statement D

The statement is 'A linear transformation preserves the operations of vector addition and scalar multiplication.' To be classified as linear, a transformation must satisfy: \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \) and \( T(c \mathbf{u}) = c T(\mathbf{u}) \). These properties show that linear transformations preserve vector addition and scalar multiplication, confirming the statement is **True**.
05

Analyze Statement E

The statement is 'The superposition principle is a physical description of a linear transformation.' The superposition principle states that the response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. In mathematical terms, this is exactly what linear transformations do, as they preserve vector addition and scalar multiplication. Hence, this statement is **True**.

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Key Concepts

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

Matrix Transformation
A matrix transformation is an essential concept in linear algebra, closely tied to the operation of matrices on vectors. To understand matrix transformation, envision a matrix as a function that takes a vector as input and produces another vector as output.
This is mathematically expressed as \( T(\mathbf{x}) = A\mathbf{x} \), where \( A \) is a matrix, \( \mathbf{x} \) is a vector, and \( T \) is the transformation.
  • Linearity Property: For a transformation to be considered linear, it must satisfy two key properties: vector addition and scalar multiplication. This demonstrates the inherent linearity in every matrix transformation.
  • Domain and Codomain: If a matrix \( A \) is \( m \times n \), then the transformation maps n-dimensional vectors to m-dimensional vectors.
Understanding these aspects enables a deeper insight into how matrix transformations function and why they are fundamental to linear transformations.
Vector Addition
Vector addition is a fundamental operation in linear algebra, playing a crucial role in linear transformations. It involves adding two vectors together to create a new vector. In mathematical terms, if you have vectors \( \mathbf{u} \) and \( \mathbf{v} \), their sum is given by
\( \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, ..., u_n + v_n) \).
  • Commutative Property: Vector addition is commutative, meaning \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \).
  • Associative Property: It's also associative, so \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \).
These properties ensure that the operation of vector addition is preserved within linear transformations. This means that applying a linear transformation to the sum of vectors gives the same result as summing the transformations applied to each vector individually.
Scalar Multiplication
Scalar multiplication involves multiplying a vector by a scalar (a single number), which scales the vector's magnitude without changing its direction. In mathematical terms, if \( c \) is a scalar and \( \mathbf{u} \) is a vector, the scalar multiplication is presented as:
\[ c \mathbf{u} = (cu_1, cu_2, ..., cu_n) \].
This operation is vital in the context of linear transformations, ensuring that scaling a vector and then transforming it yields the same outcome as transforming the vector and then scaling the result.
  • Distributive Property: Scalar multiplication respects distributive properties both with addition of vectors and multiplication of scalars: \( c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v} \).
  • Consistency in Linearity: Consistency of this process within a linear transformation is another cornerstone that qualifies transformations involving matrices as linear.
Scalar multiplication confirms one of the basic properties needed for linear transformations, maintaining the proportionality in transformations.
Superposition Principle
The superposition principle is a core concept in physics and mathematics, describing how systems respond to combinations of inputs. In the context of linear transformations, it underscores how they handle sums of vectors and scaled vectors.
This principle states that if multiple forces act on a system, the total effect is simply the sum of the individual effects, reflecting the linearity observed in transformations.
  • Additive Response: If \( T \) is a linear transformation and \( \mathbf{u} \) and \( \mathbf{v} \) are vectors, then \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \).
  • Scaling Consistency: For any scalar \( c \), \( T(c\mathbf{u}) = cT(\mathbf{u}) \). This emphasizes that transformations retain linearity even when inputs are varied in scale.
This principle is foundational, not just in linear algebra but across numerous scientific fields, explaining behaviors through linear approximations and ensuring predictable outcomes when transformations are applied.

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Most popular questions from this chapter

Construct a \(3 \times 3\) nonzero matrix \(A\) such that the vector \(\left[\begin{array}{r}{1} \\ {-2} \\ {1}\end{array}\right]\) is a solution of \(A \mathbf{x}=\mathbf{0}\) .

Suppose an economy has four sectors, Agriculture \((\mathrm{A}),\) Energy (E), Manufacturing \((\mathrm{M}),\) and Transportation (T). Sector A sells 10\(\%\) of its output to \(\mathrm{E}\) and 25\(\%\) to \(\mathrm{M}\) and retains the rest. Sector \(\mathrm{E}\) sells 30\(\%\) of its output to \(\mathrm{A}, 35 \%\) to \(\mathrm{M},\) and 25\(\%\) to \(\mathrm{T}\) and retains the rest. Sector \(\mathrm{M}\) sells 30\(\%\) of its output to \(\mathrm{A}, 15 \%\) to \(\mathrm{E},\) and 40\(\%\) to \(\mathrm{T}\) and retains the rest. Sector \(\mathrm{T}\) sells 20\(\%\) of its output to \(\mathrm{A}, 10 \%\) to \(\mathrm{E},\) and 30\(\%\) to \(\mathrm{M}\) and retains the rest. a. Construct the exchange table for this economy. b. \([\mathbf{M}]\) Find a set of equilibrium prices for the economy.

In a certain region, about 6% of a city鈥檚 population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city. In 2015, there were 10,000,000 residents in the city and 800,000 in the suburbs. Set up a difference equation that describes this situation, where \(\mathbf{x}_{0}\) is the initial population in 2015 . Then estimate the populations in the city and in the suburbs two years later, in 2017 .

Find the value(s) of \(h\) for which the vectors are linearly dependent. Justify each answer. \(\left[\begin{array}{r}{1} \\ {5} \\\ {-3}\end{array}\right],\left[\begin{array}{r}{-2} \\ {-9} \\\ {6}\end{array}\right],\left[\begin{array}{r}{3} \\ {h} \\\ {-9}\end{array}\right]\)

Each statement in Exercises 33鈥�38 is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is true, give a justification. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.) If \(\mathbf{v}_{1}, \ldots, \mathbf{v}_{4}\) are linearly independent vectors in \(\mathbb{R}^{4},\) then \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\\}\) is also linearly independent. [Hint: Think about \(x_{1} \mathbf{v}_{1}+x_{2} \mathbf{v}_{2}+x_{3} \mathbf{v}_{3}+0 \cdot \mathbf{v}_{4}=\mathbf{0} . ]\)
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