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 [latex]\begin{bmatrix} 1 & b \\ c & d \end{bmatrix}[/latex] with the standard operations is a vector space.

To show that the set of matrices of the given form is not a vector space, let [latex]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} [/latex] and [latex]B = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}[/latex] Both A and B are of the given form, but the sum of these matrices, [latex]A + B = \begin{bmatrix} 2 & 7 \\ 9 & 11 \end{bmatrix}[/latex] is not. This means that the set does not have closure under addition, so it does not satisfy the first axiom in the definition.

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

Elementary Linear Algebra

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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