When doing a matrix-times-vector multiplication over F_2, there is some
inefficiency caused by calling `VectorSpace` to create a parent for the
result. This becomes noticeable when repeatedly multiplying small
matrices.

When the dimensions are all the same, this can be improved by using the
parent of the vector also as the parent for the result. Before:
```
sage: A = random_matrix(GF(2),10^2,10^2)
sage: v0 = vector(random_matrix(GF(2),10^2,1))
sage: %timeit A*v0
2.78 µs ± 128 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops
each)
```
After:
```
sage: A = random_matrix(GF(2),10,10)
sage: v0 = vector(random_matrix(GF(2),10,1))
sage: %timeit A*v0
995 ns ± 22.9 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops
each)
```

URL: sagemath#37375
Reported by: kedlaya
Reviewer(s): Lorenz Panny

sagemathgh-37375: Speed up square matrix times vector over GF(2)
    
When doing a matrix-times-vector multiplication over F_2, there is some
inefficiency caused by calling `VectorSpace` to create a parent for the
result. This becomes noticeable when repeatedly multiplying small
matrices.

When the dimensions are all the same, this can be improved by using the
parent of the vector also as the parent for the result. Before:
```
sage: A = random_matrix(GF(2),10^2,10^2)
sage: v0 = vector(random_matrix(GF(2),10^2,1))
sage: %timeit A*v0
2.78 µs ± 128 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops
each)
```
After:
```
sage: A = random_matrix(GF(2),10,10)
sage: v0 = vector(random_matrix(GF(2),10,1))
sage: %timeit A*v0
995 ns ± 22.9 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops
each)
```
    
URL: sagemath#37375
Reported by: kedlaya
Reviewer(s): Lorenz Panny

…matrix-vector multiplication over GF(2)

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description below. -->
<!-- ^ For example, instead of "Fixes sagemath#12345" use "Introduce new method
to calculate 1 + 2". -->
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example, "Fixes sagemath#12345". -->

Fixes [sagemath#40167](sagemath#40167). This PR
fixes a bug in `Matrix_mod2_dense._matrix_times_vector_` revealed in
[sagemath#40167](sagemath#40167), where the
parent of the resulting vector was incorrectly reused from the input
vector.

The regression was introduced in [PR
sagemath#37375](sagemath#37375), which added an
optimization to speed up matrix-vector multiplication when the matrix is
square and matches the vector dimension. However, it failed to account
for edge cases where the input vector's parent is not the full ambient
vector space—for example, when working with subspaces. In this case, we
can no longer assume that the ambient space is the vector's ambient
space.

In such cases, reusing the parent leads to incorrect coercion or a
result vector with an invalid parent space. This patch introduces an
explicit check: if the vector's parent is the full space `GF(2)^n`, it
is reused; otherwise, a default parent is constructed to ensure
correctness.

### Example (correct behavior restored)

```python
sage: M = Matrix(GF(2), [[1, 1], [0, 1]])
sage: v = vector(GF(2), [0, 1])
sage: V = span([v])             # one-dimensional subspace of GF(2)^2
sage: image_basis = [M * b for b in V.basis()]
sage: image = span(image_basis)
sage: image.basis() == [(1, 1)]
True # now returns True
```

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

<!-- List all open PRs that this PR logically depends on. For example,
-->
<!-- - sagemath#12345: short description why this is a dependency -->
<!-- - sagemath#34567: ... -->

URL: sagemath#40176
Reported by: Aolong Li
Reviewer(s): Travis Scrimshaw