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Chapter 1: Q22Q (page 1)

Let \(A\) be a \(3 \times 3\) matrix with the property that the linear transformation \({\bf{x}} \mapsto A{\bf{x}}\) maps \({\mathbb{R}^3}\) into \({\mathbb{R}^3}\). Explain why transformation must be one-to-one.

Short Answer

Expert verified

The transformation \(x \mapsto Ax\) is one to one.

Step by step solution

01

Write the details of matrix \(A\)

The order of matrix \(A\) is \(3 \times 3\). The transformation for \(A\) \(x \mapsto Ax\) is linear, and it maps \({\mathbb{R}^3} \to {\mathbb{R}^3}\).

02

Recall the concept of one-to-one transformation

For any matrix \(A\) whose transformation is one-to-one, the equation \(A{\bf{x}} = 0\) cannot have any free variable. Also, all the columns of the matrix should be linearly independent.

03

Make the transformation using the concept of linear independence

As \(x \mapsto Ax\) and \({\mathbb{R}^3}\) maps to \({\mathbb{R}^3}\), \(A\) has a pivot in each of its three rows. Since \(A\) has three columns, each column must be a pivot column. It means the equation \(A{\bf{x}} = 0\) has no free variables, and all the columns of \(A\) are linearly independent.

Therefore, the transformation \(x \mapsto Ax\) is one-to-one.

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Most popular questions from this chapter

Suppose \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\) are distinct points on one line in \({\mathbb{R}^3}\). The line need not pass through the origin. Show that \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\) is linearly dependent.

Construct three different augmented matrices for linear systems whose solution set is \({x_1} = - 2,{x_2} = 1,{x_3} = 0\).

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.]

Construct a \(3 \times 3\) matrix\(A\), with nonzero entries, and a vector \(b\) in \({\mathbb{R}^3}\) such that \(b\) is not in the set spanned by the columns of\(A\).

Give a geometric description of span \(\left\{ {{v_1},{v_2}} \right\}\) for the vectors \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}8\\2\\{ - 6}\end{array}} \right]\) and \({{\mathop{\rm v}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}{12}\\3\\{ - 9}\end{array}} \right]\).
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