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Question 4.2.3: The Vector Space of All 2 × 3 Matrices Show that the set of ...

The Vector Space of All 2 × 3 Matrices

Show that the set of all 2 × 3 matrices with the operations of matrix addition and scalar multiplication is a vector space.

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If A and B are 2 × 3 matrices and c is a scalar, then A + B and cA are also 2 × 3 matrices. The set is, therefore, closed under matrix addition and scalar multiplication. Moreover, the other eight vector space axioms follow directly from Theorems 2.1 and 2.2 (see Section 2.2). So, the set is a vector space. Vectors in this space have the form

a =A = \left [ \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{matrix} \right ].

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