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Section 3.8 The Big Equivalences Theorem

Recall that the point \(\vect{v} = \oq{v_1}{v_2}{\ldots}{v_n} \in \mathbb{R}^n\) can be written as the list of coefficients using the standard basis \(\beta\) which gives the \(n \times 1\) matrix

\begin{equation*} \vect{v} =[(v_1,v_2, ..,v_n)]_{\beta} = \left[ \begin{array}{c} v_1\\v_2\\ \vdots \\v_n \end{array} \right] \end{equation*}

This allows multiplication on the left by an \(n \times n\) matrix \(\mtx{M}\) to result in a new vector \(\vect{b} \in \mathbb{R}^n\) written: \(\mtx{M}\vect{v} = \vect{b}\text{.}\)

For any \(\vect{u}=\ot{u_1}{u_2}{u_3}, \vect{v}=\ot{v_1}{v_2}{v_3}\text{,}\) \(\vect{w}=\ot{w_1}{w_2}{w_3}\) and \(\vect{a}=\ot{a_1}{a_2}{a_3} \in \mathbb{R}^3\text{,}\) with \(c_i \in \mathbb{R}\text{,}\) write the equation:

\(c_1\vect{u}+c_2\vect{v}+c_3\vect{w} = \vect{a}\) as:

  1. a linear combination: \(\left[ \begin{array}{c} \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \end{array} \right] + \underline{\hspace{.2in} } \left[ \begin{array}{c} \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \end{array} \right]+ \underline{\hspace{.2in} } \left[ \begin{array}{c} \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \end{array} \right] = \left[ \begin{array}{c} \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \end{array} \right]\)

  2. an augmented matrix: \(\left[ \begin{array}{ccccc} \underline{\hspace{.2in} } \amp \underline{\hspace{.2in} } \amp \underline{\hspace{.2in} } \amp \vdots \amp \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \amp \underline{\hspace{.2in} } \amp \underline{\hspace{.2in} } \amp \vdots \amp \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \amp \underline{\hspace{.2in} } \amp \underline{\hspace{.2in} } \amp \vdots \amp \underline{\hspace{.2in} } \end{array} \right]\)

  3. a matrix equation: \(\left[ \begin{array}{ccc} \underline{\hspace{.2in} } \amp \underline{\hspace{.2in} } \amp \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \amp \underline{\hspace{.2in} } \amp \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \amp \underline{\hspace{.2in} } \amp \underline{\hspace{.2in} } \end{array} \right]\) \(\left[ \begin{array}{c} \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \end{array} \right]\) = \(\left[ \begin{array}{c} \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \\ \underline{\hspace{.2in} } \end{array} \right]\)

Write the augmented matrix obtained by setting \(\vect{a} = \ot{a_1}{a_2}{a_3}\) equal to a linear combination of the vectors \(\{ \ot{1}{4}{2}, \ot{0}{2}{1}, \ot{-1}{0}{0}, \ot{1}{2}{3} \}\text{.}\) Determine if these vectors are linearly independent and span \(\mathbb{R}^3\text{.}\)

For a given \(r \times c\) matrix \(\mtx{A}\text{,}\) sort the following statements into two groups such that the statements in each group are equivalent to each other:

The columns of \(\mtx{A}\) are linearly independent.

The columns of \(\mtx{A}\) span \(\mathbb{R}^r\text{.}\)

There is a leading 1 in each row when \(\mtx{A}\) is row reduced.

There is a leading 1 in each column when \(\mtx{A}\) is row reduced.

Any system of equations with coefficients from \(\mtx{A}\) will have \(\leq 1\) solution.

Any system of equations with coefficients from \(\mtx{A}\) will have \(\geq 1\) solution.

The matrix equation \(\mtx{A}\vect{x}=\vect{b}\) will have \(\leq 1\) solution.

The matrix equation \(\mtx{A}\vect{x}=\vect{b}\) will have \(\geq 1\) solution.

For an \(n \times n\) matrix \(\mtx{A}\) with \(det(\mtx{A}) \neq 0\text{,}\) how can elementary operations be used to find \(\mtx{A}^{-1}\text{?}\)