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Chapter 1: Q22E (page 1)

In Exercise 22, mark each statement True or False. Justify each answer.

22. a. Every matrix transformation is a linear transformation.

b. The codomain of the transformation \({\bf{x}} \mapsto {\bf{Ax}}\) is the set of all linear combinations of the columns of \({\bf{A}}\).

c. If \({\bf{T}}:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{m}}}\) is a linear transformation and if \({\bf{c}}\) is in \({\mathbb{R}^{\bf{m}}}\), then a uniqueness is 鈥淚s c in the range of T?鈥�

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
  1. The statement is true.
  2. The statement is false.
  3. The statement is false.
  4. The statement is true.
  5. The statement is true.

Step by step solution

01

Use the properties of matrices

(a)

Suppose \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) by \(T\left( x \right) = Ax\) is a matrix transformation, where \(A\) is a \(m \times n\) matrix.

Claim: \(T\) is linear.

(i) \(T\left( {u + v} \right) = A\left( {u + v} \right)\)

\(\begin{array}{c}T\left( {u + v} \right) = Au + Av\\T\left( {u + v} \right) = T\left( u \right) + T\left( v \right)\end{array}\)

(ii) \(T\left( {cw} \right) = A\left( {cw} \right)\)

\(\begin{array}{c}T\left( {cw} \right) = cAw\\T\left( {cw} \right) = cT\left( w \right)\end{array}\)

From (i) and (ii), you have \(T\) is linear.

Hence, the given statement is true.

02

Use the definition of matrix transformation

(b)

Suppose \(A\) is a \(m \times n\) matrix. Then, the dimension of the codomain should be \(m\). However, the dimension of the set of all linear combinations of the columns of \(A\) need not be \(m\).

Hence, the given statement is false.

03

Use the definition of the range of \({\bf{T}}\)

(c)

The range of \(T\) is contained in \({\mathbb{R}^m}\). So, there always exists a \(c \in {\mathbb{R}^m}\) such that \(T\left( {ax + by} \right) \ne c\) for all scalars \(a,\,b\) and for all vectors \(x,\,y\) in \({\mathbb{R}^n}\). That is, \(c\) does not belong to the range of \(T\).

Hence, the given statement is false.

04

Use the definition of a linear transformation

(d)

Let \(T\) be linear. Then,

\(T\left( {u + v} \right) = T\left( u \right) + T\left( v \right)\)

This implies \(T\) preserves the vector addition.

Also,

\(T\left( {cw} \right) = cT\left( w \right)\)

This implies \(T\) preserves the scalar multiplication.

Here, \(c\) is scalar and \(u,\,v,\) and \(w\)are vectors in the domain of\(T\).

Hence, the given statement is true.

05

Use the superposition rule

(e)

Let \(T\) be a linear transformation. Then,

\(T\left( {{c_1}{v_1} + {c_2}{v_2} + ... + {c_n}{v_n}} \right) = {c_1}T\left( {{v_1}} \right) + {c_2}T\left( {{v_2}} \right) + ... + {c_3}T\left( {{v_n}} \right)\)

It is known as the superposition rule. Think of \({v_1},{v_2},...,{v_n}\) as signals that go into the system and \(T\left( {{v_1}} \right),\,T\left( {{v_2}} \right),...,T\left( {{v_n}} \right)\) as the responses of that system to the signals.

Hence, the given statement is true.

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Most popular questions from this chapter

Question:Let A be the n x n matrix with 0's on the main diagonal, and 1's everywhere else. For an arbitrary vector b鈫�in 鈻�n, solve the linear system Ax鈫�=b鈬赌, expressing the components x1,.......,xnof x鈫�in terms of the components of b鈬赌. See Exercise 69 for the case n=3 .

In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar statement appears, or refer to a de铿乶ition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text.

23.

a. Every elementary row operation is reversible.

b. A \(5 \times 6\)matrix has six rows.

c. The solution set of a linear system involving variables \({x_1},\,{x_2},\,{x_3},........,{x_n}\)is a list of numbers \(\left( {{s_1},\, {s_2},\,{s_3},........,{s_n}} \right)\) that makes each equation in the system a true statement when the values \ ({s_1},\, {s_2},\, {s_3},........,{s_n}\) are substituted for \({x_1},\,{x_2},\,{x_3},........,{x_n}\), respectively.

d. Two fundamental questions about a linear system involve existence and uniqueness.

Let \(T:{\mathbb{R}^3} \to {\mathbb{R}^3}\) be the linear transformation that reflects each vector through the plane \({x_{\bf{2}}} = 0\). That is, \(T\left( {{x_1},{x_2},{x_3}} \right) = \left( {{x_1}, - {x_2},{x_3}} \right)\). Find the standard matrix of \(T\).

Let \(A = \left[ {\begin{array}{*{20}{c}}1&0&{ - 4}\\0&3&{ - 2}\\{ - 2}&6&3\end{array}} \right]\) and \(b = \left[ {\begin{array}{*{20}{c}}4\\1\\{ - 4}\end{array}} \right]\). Denote the columns of \(A\) by \({{\mathop{\rm a}\nolimits} _1},{a_2},{a_3}\) and let \(W = {\mathop{\rm Span}\nolimits} \left\{ {{a_1},{a_2},{a_3}} \right\}\).

  1. Is \(b\) in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)? How many vectors are in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)?
  2. Is \(b\) in \(W\)? How many vectors are in W.
  3. Show that \({a_1}\) is in W.[Hint: Row operations are unnecessary.]

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be an invertible linear transformation, and let Sand U be functions from \({\mathbb{R}^n}\) into \({\mathbb{R}^n}\) such that \(S\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) and \(\)\(U\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\). Show that \(U\left( v \right) = S\left( v \right)\) for all v in \({\mathbb{R}^n}\). This will show that Thas a unique inverse, as asserted in theorem 9. (Hint: Given any v in \({\mathbb{R}^n}\), we can write \({\mathop{\rm v}\nolimits} = T\left( {\mathop{\rm x}\nolimits} \right)\) for some x. Why? Compute \(S\left( {\mathop{\rm v}\nolimits} \right)\) and \(U\left( {\mathop{\rm v}\nolimits} \right)\)).
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