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Chapter 1: Q29E (page 1)

Question: In Exercises 29 and 30, describe the possible echelon forms of the standard matrix for a linear transformation \(T\). Use the notation of Example 1 in section 1.2.

29. \(T:{\mathbb{R}^3} \to {\mathbb{R}^4}\) is one-to-one.

Short Answer

Expert verified

The possible echelon form of the standard matrix is

\(\left[ {\begin{array}{*{20}{c}} \square & * & * \\ 0&\square & * \\ 0&0&\square \\ 0&0&0 \end{array}} \right]\) .

Step by step solution

01

The notation of example 1 for matrices in echelon form

In example 1, the following matrices are in echelon form. The leading entries \(\left( \square \right)\) may have any nonzero value; the starred entries \(\left( * \right)\) may have any value (including zero).

\(\left[ {\begin{array}{*{20}{c}} \square & * & * & * \\ 0&\square & * & * \\ 0&0&0&0 \\ 0&0&0&0 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 0&\square & * & * & * & * & * & * & * & * \\ 0&0&0&\square & * & * & * & * & * & * \\ 0&0&0&0&\square & * & * & * & * & * \\ 0&0&0&0&0&\square & * & * & * & * \\ 0&0&0&0&0&0&0&0&\square & * \end{array}} \right]\)

02

Determine the possible echelon form of the standard matrix

Theorem 12states that let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation, and let \(A\) be the standard matrix \(T\) then:

  1. \(T\)maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\) if and only if the columns of \(A\) span\({\mathbb{R}^m}\).
  2. \(T\)is one-to-one if and only if the columns of \(A\) are linearly independent.

The columns of \(A\) must be linearly independent, and so the equation \(Ax = 0\) contains no free variable according to theorem 12. Therefore, \(A\) must have a pivot column in each column.

Use leading entries \(\left( \square \right)\) and starred entries \(\left( * \right)\) to write the possible echelon form of the matrix.

\(\left[ {\begin{array}{*{20}{c}} \square & * & * \\ 0&\square & * \\ 0&0&\square \\ 0&0&0 \end{array}} \right]\) .

The shape of \(A\) prevents \(T\) from being one-to-one.

Thus, the possible echelon form of the standard matrix is \(\left[ {\begin{array}{*{20}{c}} \square & * & * \\ 0&\square & * \\ 0&0&\square \\ 0&0&0 \end{array}} \right]\) . .

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Most popular questions from this chapter

Suppose Tand Ssatisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that Sis a linear transformation. (Hint: Given u, v in \({\mathbb{R}^n}\), let \({\mathop{\rm x}\nolimits} = S\left( {\mathop{\rm u}\nolimits} \right),{\mathop{\rm y}\nolimits} = S\left( {\mathop{\rm v}\nolimits} \right)\). Then \(T\left( {\mathop{\rm x}\nolimits} \right) = {\mathop{\rm u}\nolimits} \), \(T\left( {\mathop{\rm y}\nolimits} \right) = {\mathop{\rm v}\nolimits} \). Why? Apply Sto both sides of the equation \(T\left( {\mathop{\rm x}\nolimits} \right) + T\left( {\mathop{\rm y}\nolimits} \right) = T\left( {{\mathop{\rm x}\nolimits} + y} \right)\). Also, consider \(T\left( {cx} \right) = cT\left( x \right)\).)

Consider a dynamical system x鈫�(t+1)=Ax鈫�(t) with two components. The accompanying sketch shows the initial state vector x鈫�0and two eigen vectors 蠀1鈫�鈥�鈥�and鈥�鈥�蠀2鈫� of A (with eigen values 位1鈫�and位2鈫� respectively). For the given values of 位1鈫�and位2鈫�, draw a rough trajectory. Consider the future and the past of the system.

位1鈫�=1,位2鈫�=0.9

In Exercises 11 and 12, determine if \({\rm{b}}\) is a linear combination of \({{\mathop{\rm a}\nolimits} _1},{a_2}\) and \({a_3}\).

11.\({a_1} = \left[ {\begin{array}{*{20}{c}}1\\{ - 2}\\0\end{array}} \right],{a_2} = \left[ {\begin{array}{*{20}{c}}0\\1\\2\end{array}} \right],{a_3} = \left[ {\begin{array}{*{20}{c}}5\\{ - 6}\\8\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\\6\end{array}} \right]\)

Let a and b represent real numbers. Describe the possible solution sets of the (linear) equation \(ax = b\). (Hint:The number of solutions depends upon a and b.)

In Exercises 13 and 14, determine if \(b\) is a linear combination of the vectors formed from the columns of the matrix \(A\).

13. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 4}&2\\0&3&5\\{ - 2}&8&{ - 4}\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}3\\{ - 7}\\{ - 3}\end{array}} \right]\)
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