Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| group |
✓ |
The $2 \times 2$ real matrices form a group under matrix addition, because the zero matrix is an identity and $-A$ is the inverse of $A$.
|
| ring |
✓ |
The $2 \times 2$ real matrices form a ring, because matrix multiplication is associative, has the identity matrix $I$ as its identity, and distributes over addition.
|
| vector space |
✓ |
The $2 \times 2$ real matrices form a real vector space of dimension $4$, with entrywise addition and scalar multiplication by real numbers.
|
| module |
✓ |
The $2 \times 2$ real matrices form a module over $\reals$, because they form a vector space over $\reals$.
|
| algebra (over a field) |
✓ |
The $2 \times 2$ real matrices form an algebra over $\reals$, because matrix multiplication is bilinear: $(a A) B = a (A B) = A (a B)$, and matrix multiplication distributes over addition.
|
| field |
✗ |
The $2 \times 2$ real matrices do not form a field, because matrix multiplication is not commutative, and nonzero singular matrices, such as \(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\), have no inverse.
|
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