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Chapter 1: Q12E (page 39)
Question: There exists a 2x2 matrix such that.
Answer:
True, there exist a2x2 matrix A such that
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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 \({{\mathop{\rm v}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}{12}\\3\\{ - 9}\end{array}} \right]\).
In Exercises 11 and 12, determine if \({\rm{b}}\) is a linear combination of \({{\mathop{\rm a}\nolimits} _1},{a_2}\) and \({a_3}\).
12.
If Ais an \(n \times n\) matrix and the transformation \({\bf{x}}| \to A{\bf{x}}\) is one-to-one, what else can you say about this transformation? Justify your answer.
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.
Consider the dynamical system .
Sketch a phase portrait of this system for the given values of :
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