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Chapter 1: Q21E (page 1)

In Exercise 21, mark each statement True or False. Justify each answer.\(\) \(\)

21. a. A linear transformation is a special type of function.

b. If \({\bf{A}}\) is a \({\bf{3}} \times {\bf{5}}\)matrix and \({\bf{T}}\) is a transformation defined by\({\bf{T}}\left( {\bf{x}} \right) = {\bf{Ax}}\), then the domain of \({\bf{T}}\) is\({\mathbb{R}^{\bf{3}}}\).

c. If \({\bf{A}}\) is an \({\bf{m}} \times {\bf{n}}\) matrix, then the range of the transformation \({\bf{x}} \mapsto {\bf{Ax}}\) is\({\mathbb{R}^{\bf{m}}}\).

\(\) d. Every linear transformation is a matrix transformation.

e. A transformation \({\bf{T}}\) is linear if and only if \({\bf{T}}\left( {{{\bf{c}}_{\bf{1}}}{{\bf{v}}_{\bf{1}}} + {{\bf{c}}_{\bf{2}}}{{\bf{v}}_{\bf{2}}}} \right) = {{\bf{c}}_{\bf{1}}}{\bf{T}}\left( {{{\bf{v}}_{\bf{1}}}} \right) + {{\bf{c}}_{\bf{2}}}{\bf{T}}\left( {{{\bf{v}}_{\bf{2}}}} \right)\) for all \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) in the domain of \({\bf{T}}\) and for all scalars \({{\bf{c}}_{\bf{1}}}\) and\({{\bf{c}}_{\bf{2}}}\).\[\]

Short Answer

Expert verified
  1. The given statement is true
  2. The given statement is false.
  3. The given statement is false.
  4. The given statement is false.
  5. The given statement is true.

Step by step solution

01

Use the definition of linear transformation

(a)

Note that every linear transformation is a function. However, not every function is a linear transformation. When a function satisfies the following axioms, only then will it be linear.

i.e., (i) \(T\left( {u + v} \right) = T\left( u \right) + T\left( v \right)\)

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

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

The above axioms make the linear transformation a special function.

Hence, the given statement is true.

02

Use the fact of matrix \({\bf{A}}\)

(b)

Given, \(A\) is a \(3 \times 5\) matrix. This implies \(A\) contains 5 column vectors. Since \(T\left( x \right) = Ax\), the domain of \(T\)is \({\mathbb{R}^5}\).

Thus, the given statement is false.

03

Use the definition of matrix transformation

(c)

The given matrix transformation is \(x \mapsto Ax\) and \(A\) is a \(m \times n\) matrix. This implies its domain is \({\mathbb{R}^n}\) and its codomain is \({\mathbb{R}^m}\). Note that the range of the transformation is the set of all linear combinations of the column vectors in \(A\). This implies the range is contained in \({\mathbb{R}^m}\) but not exactly\({\mathbb{R}^m}\). This is possible if and only if the transformation is onto.

Hence, the given statement is false.

04

Use the fact of a linear transformation 

(d)

Every matrix transformation is a linear transformation. However, the converse need not be true. Hence, every linear transformation is not a matrix transformation.

Thus, the given statement is false.

05

Use the definition of a linear transformation

(e)

Suppose \(T\) is a linear transformation. This implies

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

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

Consider,

\(\begin{aligned}{c}T\left( {{c_1}{v_1} + {c_2}{v_2}} \right) &= T\left( {{c_1}{v_1}} \right) + T\left( {{c_2}{v_2}} \right)\\ &= {c_1}T\left( {{v_1}} \right) + {c_2}T\left( {{v_2}} \right)\end{aligned}\)

Suppose \(T\left( {{c_1}{v_1} + {c_2}{v_2}} \right) = {c_1}T\left( {{v_1}} \right) + {c_2}T\left( {{v_2}} \right)\)

This implies \(T\left( {{c_1}{v_1} + {c_2}{v_2}} \right) = T\left( {{c_1}{v_1}} \right) + T\left( {{c_2}{v_2}} \right)\).

Thus, \(T\) is linear.

Hence, the given statement is true.

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Most popular questions from this chapter

Mark each statement True or False. Justify each answer.

a. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.

b. The row reduction algorithm applies only to augmented matrices for a linear system.

c. A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.

d. Finding a parametric description of the solution set of a linear system is the same as solving the system.

e. If one row in an echelon form of an augmented matrix is \(\left( {\begin{array}{*{20}{c}}0&0&0&5&0\end{array}} \right)\), then the associated linear system is inconsistent.

In Exercises 23-26, describe the possible echelon forms of the matrix. Use the notation of Example 1 in Section 1.2

23. \(A\) is a \(3 \times 3\) matrix with linearly independent columns.

Each statement in Exercises 33-38 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.)

33. If \({{\mathop{\rm v}\nolimits} _1},...,{v_4}\) are in \({\mathbb{R}^4}\) and \({{\mathop{\rm v}\nolimits} _3} = 2{{\mathop{\rm v}\nolimits} _1} + {v_2}\), then \(\left\{ {{v_1},{v_2},{v_3},{v_4}} \right\}\) is linearly dependent.

Use Theorem 7 in section 1.7 to explain why the columns of the matrix Aare linearly independent.

\(A = \left( {\begin{aligned}{*{20}{c}}1&0&0&0\\2&5&0&0\\3&6&8&0\\4&7&9&{10}\end{aligned}} \right)\)

As in Exercise 15, describe the solutions of the following system in parametric vector form, and provide a geometric comparison with the solution set in Exercise 6.

\(\begin{array}{c}{x_1} + 3{x_2} - 5{x_3} = 4\\{x_1} + 4{x_2} - 8{x_3} = 7\\ - 3{x_1} - 7{x_2} + 9{x_3} = - 6\end{array}\)
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