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In Exercises 23 and 24, mark each statement True or False. Justify each answer.

24.

a. Not every linear transformation from\({\mathbb{R}^n}\)to\({\mathbb{R}^m}\)is a matrix transformation.

b. The columns of the standard matrix for a linear transformation from\({\mathbb{R}^n}\)to\({\mathbb{R}^m}\)are the images of the columns of the\(n \times n\)identity matrix.

c. The standard matrix of a linear transformation from\({\mathbb{R}^2}\)to\({\mathbb{R}^2}\)that reflects points through the horizontal axis, the vertical axis, or the origin has the form\(\left[ {\begin{array}{*{20}{c}}a&0\\0&d\end{array}} \right]\), where\(a\)and\(d\)are\( \pm 1\).

d. A mapping\(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\)is one-to-one if each vector in\({\mathbb{R}^n}\)maps onto a unique vector in\({\mathbb{R}^m}\).

e. If\(A\)is a\(3 \times 2\)matrix, then the transformation\(x \mapsto Ax\)cannot map\({\mathbb{R}^2}\)onto\({\mathbb{R}^3}\).

Short Answer

Expert verified

a. The given statement is false.

b. The given statement is true.

c. The given statement is true.

d. The given statement is false.

e. The given statement is true.

Step by step solution

01

Determine whether the given statement is true or false

(a)

Every linear transformation from\({\mathbb{R}^n}\)to\({\mathbb{R}^m}\)can be viewed as a matrix transformation.

Thus, the given statement (a) is false.

02

Determine whether the given statement is true or false

(b)

Theorem 10 states that let\(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\)be a linear transformation, then there exists a unique matrix\(A\)such that\(T\left( x \right) = Ax\)for all\(x\)in\({\mathbb{R}^n}\). \(A\)is the\(m \times n\)matrix whose\[j{\mathop{\rm th}\nolimits} \]column of the identity matrix in\({\mathbb{R}^n}\):\(A = \left[ {\begin{array}{*{20}{c}}{T\left( {{e_1}} \right)}&{...}&{T\left( {{e_n}} \right)}\end{array}} \right]\).

Thus, the given statement (b) is true.

03

Determine whether the given statement is true or false

(c)

The standard matrix of a linear transformation from\({\mathbb{R}^2}\)to\({\mathbb{R}^2}\)that reflects points through the horizontal axis, the vertical axis or the origin has the form\(\left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right]\).

Thus, the given statement (c) is true.

04

Determine whether the given statement is true or false

(d)

A transformation\(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\)is said to be one-to-one\({\mathbb{R}^m}\)if each\[{\mathop{\rm b}\nolimits} \]in\({\mathbb{R}^m}\)is the image of at most one\(x\)in\({\mathbb{R}^n}\).

Thus, the given statement (d) is false.

05

Determine whether the given statement is true or false

(e)

Consider that\(A\)is a\(3 \times 2\)matrix; the columns of\(A\)span\({\mathbb{R}^3}\)if and only if\(A\)has 3 pivot columns. Since\(A\)has only two columns, so the columns of\(A\)do not span\({\mathbb{R}^3}\), and the linear transformation does not map\({\mathbb{R}^2}\)onto\({\mathbb{R}^3}\).

Thus, the given statement (e) is true.

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Most popular questions from this chapter

If A is a \({\bf{4}} \times {\bf{3}}\) matrix, what is the largest possible dimension of the row space of A? If Ais a \({\bf{3}} \times {\bf{4}}\) matrix, what is the largest possible dimension of the row space of A? Explain.

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

22. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 1}&{ - 13}&{ - 12.2}&{ - 1.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

Define by \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 1 \right)}\end{array}} \right)\). For instance, if \({\mathop{\rm p}\nolimits} \left( t \right) = 3 + 5t + 7{t^2}\), then \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}3\\{15}\end{array}} \right)\).

  1. Show that \(T\) is a linear transformation. (Hint: For arbitrary polynomials p, q in \({{\mathop{\rm P}\nolimits} _2}\), compute \(T\left( {{\mathop{\rm p}\nolimits} + {\mathop{\rm q}\nolimits} } \right)\) and \(T\left( {c{\mathop{\rm p}\nolimits} } \right)\).)
  2. Find a polynomial p in \({{\mathop{\rm P}\nolimits} _2}\) that spans the kernel of \(T\), and describe the range of \(T\).

Suppose a nonhomogeneous system of nine linear equations in ten unknowns has a solution for all possible constants on the right sides of the equations. Is it possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other? Discuss.

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

16. If \(A\) is an \(m \times n\) matrix of rank\(r\), then a rank factorization of \(A\) is an equation of the form \(A = CR\), where \(C\) is an \(m \times r\) matrix of rank\(r\) and \(R\) is an \(r \times n\) matrix of rank \(r\). Such a factorization always exists (Exercise 38 in Section 4.6). Given any two \(m \times n\) matrices \(A\) and \(B\), use rank factorizations of \(A\) and \(B\) to prove that rank\(\left( {A + B} \right) \le {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\).

(Hint: Write \(A + B\) as the product of two partitioned matrices.)

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