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Question Appendix.10: Using a Counterexample in Linear Algebra Use a counterexampl...

Using a Counterexample in Linear Algebra

Use a counterexample to show that the statement is false.

The set of all 2 × 2 matrices of the form

\begin{bmatrix} 1 & b \\ c & d \end{bmatrix}

with the standard operations is a vector space.

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To show that the set of matrices of the given form is not a vector space, let

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}   and  B = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}

Both A and B are of the given form, but the sum of these matrices,

A + B = \begin{bmatrix} 2 & 7 \\ 9 & 11 \end{bmatrix}

is not. This means that the set does not have closure under addition, so it does not satisfy the first axiom in the definition.

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