More updates to the Getting Started page

@WalterMadelim did some edits, did this cover all the questions you had?

Author: ChrisRackauckas

docs/src/tutorials/linear.md

@@ -2,8 +2,9 @@
 
 A linear system $$Au=b$$ is specified by defining an `AbstractMatrix` or `AbstractSciMLOperator`.
 For the sake of simplicity, this tutorial will start by only showcasing concrete matrices.
+And specifically, we will start by using the basic Julia `Matrix` type.
 
-The following defines a matrix and a `LinearProblem` which is subsequently solved
+The following defines a `Matrix` and a `LinearProblem` which is subsequently solved
 by the default linear solver.
 
 ```@example linsys1
@@ -57,15 +58,33 @@ sol = solve(prob, KrylovJL_GMRES()) # Choosing algorithms is done the same way
 sol.u
 ```
 
-Similerly structure matrix types, like banded matrices, can be provided using special matrix
+Similarly structure matrix types, like banded matrices, can be provided using special matrix
 types. While any `AbstractMatrix` type should be compatible via the general Julia interfaces,
-LinearSolve.jl specifically tests with the following cases:
-
-* [BandedMatrices.jl](https://github.com/JuliaLinearAlgebra/BandedMatrices.jl)
-* [BlockDiagonals.jl](https://github.com/JuliaArrays/BlockDiagonals.jl)
-* [CUDA.jl](https://cuda.juliagpu.org/stable/) (CUDA GPU-based dense and sparse matrices)
-* [FastAlmostBandedMatrices.jl](https://github.com/SciML/FastAlmostBandedMatrices.jl)
-* [Metal.jl](https://metal.juliagpu.org/stable/) (Apple M-series GPU-based dense matrices)
+LinearSolve.jl specifically tests with the following cases:		
+
+* [LinearAlgebra.jl](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/)
+  * Symmetric
+  * Hermitian
+  * UpperTriangular
+  * UnitUpperTriangular
+  * LowerTriangular
+  * UnitLowerTriangular
+  * SymTridiagonal
+  * Tridiagonal
+  * Bidiagonal
+  * Diagonal
+* [BandedMatrices.jl](https://github.com/JuliaLinearAlgebra/BandedMatrices.jl) `BandedMatrix`
+* [BlockDiagonals.jl](https://github.com/JuliaArrays/BlockDiagonals.jl) `BlockDiagonal`
+* [CUDA.jl](https://cuda.juliagpu.org/stable/) (CUDA GPU-based dense and sparse matrices) `CuArray` (`GPUArray`)
+* [FastAlmostBandedMatrices.jl](https://github.com/SciML/FastAlmostBandedMatrices.jl) `FastAlmostBandedMatrix`
+* [Metal.jl](https://metal.juliagpu.org/stable/) (Apple M-series GPU-based dense matrices) `MetalArray`
+
+!!! note
+
+  Choosing the most specific matrix structure that matches your specific system will give you the most performance.
+  Thus if your matrix is symmetric, specifically building with `Symmetric(A)` will be faster than simply using `A`,
+  and will generally lead to better automatic linear solver choices. Note that you can also choose the type for `b`,
+  but generally a dense vector will be the fastest here and many solvers will not support a sparse `b`.
 
 ## Using Matrix-Free Operators