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Chapter 1: Q23E (page 1)
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.
The possible echelon form of the \(3 \times 3\) matrix is .
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Let \(T:{\mathbb{R}^3} \to {\mathbb{R}^3}\) be the linear transformation that reflects each vector through the plane \({x_{\bf{2}}} = 0\). That is, \(T\left( {{x_1},{x_2},{x_3}} \right) = \left( {{x_1}, - {x_2},{x_3}} \right)\). Find the standard matrix of \(T\).
Suppose the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the coefficients \(c\) and \(d\)? Justify your answer.
27. \(\begin{array}{l}{x_1} + 3{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)
Question: If A is a non-zero matrix of the form, then the rank of A must be 2.
Question: Determine whether the statements that follow are true or false, and justify your answer.
14: rank.
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\(A\).
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