Remove LazyArrays.jl dependency

[ChrisRackauckas-Claude]

Summary

This PR removes the LazyArrays.jl dependency from LinearSolve.jl, simplifying the codebase as suggested.

Changes

• Removed LazyArrays from Project.toml dependencies (both main dependency and compat entry)
• Replaced BroadcastArray(@~ .-(λ .* tu)) with simple broadcast .-(λ .* tu) in src/adjoint.jl:87
• Removed LazyArrays import from src/LinearSolve.jl
• Updated tests in test/adjoint.jl to check for AbstractMatrix instead of BroadcastArray
• Removed LazyArrays import from test file

Rationale

The LazyArrays dependency was only used in one place (src/adjoint.jl:87) to create a lazy broadcast array. The regular Julia broadcast operation is sufficient for this use case and doesn't require the additional dependency. This simplification reduces the dependency footprint of the package without any loss of functionality.

Test Results

All adjoint tests pass successfully with the simplified implementation.

🤖 Generated with Claude Code

[ChrisRackauckas]

@avik-pal I assume this is overkill if Reactant is used? Because it brings in some unwanted dependencies so we kind of need to get rid of it.

[ChrisRackauckas]

Or we can make it an extension if it's a very important optimization that isn't covered by Reactant.

[avik-pal]

@ChrisRackauckas can you remind me what particular optimization this was added for? If it was for determining matrix multiplication ordering, that doesn't exist in EnzymeJAX but also like a 1 hr thing to add (atleast the greedy version).

[ChrisRackauckas]

You added it 😅. I think it was to batch backwards passes, so instead of triangular solve on vectors it would batch it to triangular solve back a matrix.

[avik-pal]

It was fa0cd34 (not me)

From discourse it seems to be the gemm ordering (I have a half finished PR for this in reactant).