91Ó°ÊÓ

Mark each of the following True or False. a. Every linear transformation is a function. b. Every function mapping \(\mathbb{R}^{n}\) into \(\mathbb{R}^{m}\) is a linear transformation. c. Composition of linear transformations corresponds to multiplication of their standard matrix representations. d. Function composition is associative. e. An invertible linear transformation mapping \(\mathbb{R}^{n}\) into itself has a unique inverse. f. The same matrix may be the standard matrix representation for several different linear transformations. g. A linear transformation having an \(m \times n\) matrix as standard matrix representation maps \(\mathbb{R}^{n}\) into \(\mathbb{R}^{m}\). h. If \(T\) and \(T^{\prime}\) are different linear transformations mapping \(\mathbb{R}^{n}\) into \(\mathbb{R}^{m}\), then we may have \(T\left(\mathbf{e}_{i}\right)=T^{\prime}\left(\mathbf{e}_{)}\right)\)for some standard basis vector \(\mathrm{e}_{i}\) of \(\mathbb{R}^{n}\). i. If \(T\) and \(T^{\prime}\) are different linear transformations mapping \(\mathbb{R}^{n}\) into \(\mathbb{R}^{r}\), then we may have \(T\left(\mathbf{e}_{i}\right)=T^{\prime}\left(\mathbf{e}_{i}\right)\) for all standard basis vectors \(\mathfrak{e}_{i}\) of \(\mathbb{R}^{n}\). j. If \(B=\left\\{\mathbf{b}_{1}, \mathbf{b}_{2}, \ldots, \mathbf{b}_{n}\right\\}\) is a basis for \(\mathbb{R}^{n}\) and \(T\) and \(T^{\prime}\) are linear transformations mapping \(\mathbb{R}^{n}\) into \(\mathbb{R}^{m}\), then \(T(\mathbf{x})=T^{\prime}(\mathbf{x})\) for all \(\mathbf{x} \in \mathbb{R}^{n}\) if and only if \(T\left(\mathbf{b}_{i}\right)=T^{\prime}\left(\mathbf{b}_{i}\right)\) for \(i=1,2, \ldots, n .\)

Step by step solution

01

Analyzing Statement a

Statement a: "Every linear transformation is a function." Explanation: A linear transformation is defined as a function between two vector spaces that preserves vector addition and scalar multiplication. Therefore, statement a is True.

02

Analyzing Statement b

Statement b: "Every function mapping \( \mathbb{R}^{n} \to \mathbb{R}^{m} \) is a linear transformation."Explanation: Not every function between these spaces is linear; a function must satisfy specific linearity conditions (additivity and homogeneity) to be considered a linear transformation. Therefore, statement b is False.

03

Analyzing Statement c

Statement c: "Composition of linear transformations corresponds to multiplication of their standard matrix representations." Explanation: The composition of linear transformations indeed corresponds to the multiplication of their corresponding matrices. Therefore, statement c is True.

04

Analyzing Statement d

Statement d: "Function composition is associative." Explanation: The associativity of function composition is a well-known property in mathematics. Therefore, statement d is True.

05

Analyzing Statement e

Statement e: "An invertible linear transformation mapping \( \mathbb{R}^{n} \) into itself has a unique inverse."Explanation: An invertible linear transformation has one and only one inverse that also maps \( \mathbb{R}^{n} \) into itself. Therefore, statement e is True.

06

Analyzing Statement f

Statement f: "The same matrix may be the standard matrix representation for several different linear transformations."Explanation: If a matrix is fixed, it corresponds uniquely to a particular linear transformation from \( \mathbb{R}^{n} \) to \( \mathbb{R}^{m} \). Therefore, statement f is False.

07

Analyzing Statement g

Statement g: "A linear transformation having an m x n matrix as standard matrix representation maps \( \mathbb{R}^{n} \) into \( \mathbb{R}^{m} \)."Explanation: The dimensions of the matrix do indicate the mapping from \( \mathbb{R}^{n} \) to \( \mathbb{R}^{m} \). Therefore, statement g is True.

08

Analyzing Statement h

Statement h: "If \( T \) and \( T' \) are different linear transformations mapping \( \mathbb{R}^{n} \) into \( \mathbb{R}^{m} \), then we may have \( T(\mathbf{e}_{i}) = T'(\mathbf{e}_{i}) \) for some standard basis vector \( \mathbf{e}_{i} \) of \( \mathbb{R}^{n} \)."Explanation: Different transformations might agree on some standard basis vectors. Therefore, statement h is True.

09

Analyzing Statement i

Statement i: "If \( T \) and \( T' \) are different linear transformations mapping \( \mathbb{R}^{n} \) into \( \mathbb{R}^{r} \), then we may have \( T(\mathbf{e}_{i}) = T'(\mathbf{e}_{i}) \) for all standard basis vectors \( \mathbf{e}_{i} \) of \( \mathbb{R}^{n} \)."Explanation: If two linear transformations are different, they cannot have the same image on all basis vectors. Therefore, statement i is False.

10

Analyzing Statement j

Statement j: "If \( B = \{ \mathbf{b}_{1}, \mathbf{b}_{2}, \ldots, \mathbf{b}_{n} \} \) is a basis for \( \mathbb{R}^{n} \) and \( T \) and \( T' \) are linear transformations mapping \( \mathbb{R}^{n} \) into \( \mathbb{R}^{m} \), then \( T(\mathbf{x})=T'(\mathbf{x}) \) for all \( \mathbf{x} \in \mathbb{R}^{n} \) if and only if \( T(\mathbf{b}_{i})=T'(\mathbf{b}_{i}) \) for \( i=1,2, \ldots, n. \)"Explanation: If two linear transformations produce the same result for each vector in the basis, then they produce the same result for all vectors in \( \mathbb{R}^{n} \). Therefore, statement j is True.

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Key Concepts

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

Function Composition

Function composition means combining two functions to create a new function. In the world of linear transformations, function composition is key. When we say the composition of two linear transformations, we refer to applying one transformation after the other. This action is similar to running commands one after another and creating a chain reaction.

Function composition is known for being associative. This means the order in which operations are grouped does not affect the outcome. To illustrate,

• If you have functions \( f \), \( g \), and \( h \), the composition \( f \circ (g \circ h) \) gives the same result as \( (f \circ g) \circ h \).

This associative property helps mathematicians and students simplify complex operations into more manageable sequences, ensuring consistency in results.

Matrix Multiplication

Matrix multiplication is a method to find the product of two matrices, fundamental in the study of linear transformations. When you compose two linear transformations, you are essentially multiplying their matrices. The product of two matrices represents the transformation resulting from applying both initial transformations.Matrix multiplication is not like regular arithmetic multiplication:

• It's not commutative: \( AB eq BA \) in general.
• The rows of the first matrix are multiplied by the columns of the second matrix.
• It is associative: \( (AB)C = A(BC) \).

This associative property aligns with function composition, streamlining the analysis of complex transformations into logical steps. This connection between matrix multiplication and linear transformations helps bridge concrete operations with abstract concepts.

Vector Spaces

Vector spaces are fundamental in understanding linear transformations. A vector space is a collection of vectors that can be scaled and added together to stay within the same space. Every linear transformation maps vectors from one vector space to another and needs two key properties:

• Additivity: \( T(u + v) = T(u) + T(v) \)
• Homogeneity: \( T(cv) = cT(v) \)

These properties ensure that the structure of the vector space is preserved, meaning you can count on transformations to behave predictably. Linear transformations serve as the bridge and help translate outcomes between two vector spaces.

Invertible Transformations

Invertible transformations are a special type of linear transformation. They are significant because they allow us to 'reverse' the effect of a transformation. When a linear transformation is invertible, there exists another transformation that undoes its effect.For a transformation \( T \) to be invertible, the following conditions must be met:

• The transformation is bijective, meaning it is both injective (one-to-one) and surjective (onto).
• The determinant of its matrix representation is non-zero.

If a linear transformation \( T \) maps \( \mathbb{R}^n \) into itself and is invertible, it has a unique inverse that will reverse the transformation's effects. This uniqueness is a crucial property as it ensures predictability and reliability within mathematical operations and problem-solving.