diff --git a/src/Grassmann.jl b/src/Grassmann.jl
index 6f4e0d5..8299b82 100644
--- a/src/Grassmann.jl
+++ b/src/Grassmann.jl
@@ -8,6 +8,7 @@ using DirectSum, AbstractTensors, Requires
 
 export ⊕, ℝ, @V_str, @S_str, @D_str, Manifold, SubManifold, Signature, DiagonalForm, value
 export @basis, @basis_str, @dualbasis, @dualbasis_str, @mixedbasis, @mixedbasis_str, Λ
+export ℝ0, ℝ1, ℝ2, ℝ3, ℝ4, ℝ5, ℝ6, ℝ7, ℝ8, ℝ9
 
 import Base: @pure, print, show, getindex, setindex!, promote_rule, ==, convert, ndims
 import DirectSum: hasinf, hasorigin, dyadmode, dual, value, V0, ⊕, pre, vsn
diff --git a/src/algebra.jl b/src/algebra.jl
index 43e4a8b..46fb7cb 100644
--- a/src/algebra.jl
+++ b/src/algebra.jl
@@ -346,13 +346,15 @@ Exterior product as defined by the anti-symmetric quotient Λ≡⊗/~
 @inline ∧(a::X,b::Y) where {X<:TensorAlgebra,Y<:TensorAlgebra} = interop(∧,a,b)
 @inline ∧(a::TensorAlgebra{V},b::UniformScaling{T}) where {V,T<:Field} = a∧V(b)
 @inline ∧(a::UniformScaling{T},b::TensorAlgebra{V}) where {V,T<:Field} = V(a)∧b
-@generated ∧(t::T) where T<:SVector = Expr(:call,:∧,[:(t[$k]) for k ∈ 1:length(t)]...)
-@generated ∧(t::T) where T<:SizedVector = Expr(:call,:∧,[:(t[$k]) for k ∈ 1:length(t)]...)
+@generated ∧(t::T) where T<:SVector{N} where N = wedges([:(t[$i]) for i ∈ 1:N])
+@generated ∧(t::T) where T<:SizedVector{N} where N = wedges([:(t[$i]) for i ∈ 1:N])
 ∧(::SVector{0,<:Chain{V}}) where V = one(V) # ∧() = 1
 ∧(::SizedVector{0,<:Chain{V}}) where V = one(V)
 ∧(t::Chain{V,1,<:Chain} where V) = ∧(value(t))
 ∧(a::X,b::Y,c::Z...) where {X<:TensorAlgebra,Y<:TensorAlgebra,Z<:TensorAlgebra} = ∧(a∧b,c...)
 
+wedges(x,i=length(x)-1) = i ≠ 0 ? Expr(:call,:∧,wedges(x,i-1),x[1+i]) : x[1+i]
+
 export ∧, ∨, ⊗
 
 @pure function ∧(a::SubManifold{V},b::SubManifold{V}) where V
diff --git a/src/composite.jl b/src/composite.jl
index eab5029..a85a41b 100644
--- a/src/composite.jl
+++ b/src/composite.jl
@@ -343,12 +343,38 @@ function Cramer(N::Int,j=0)
 end
 
 @generated function Base.:\(t::SVector{M,<:Chain{V,1}},v::Chain{V,1}) where {M,V}
-    N = ndims(V)-1 # paste this into the REPL for faster eval
-    x,y,xy = Grassmann.Cramer(N)
+    W = M≠ndims(V) ? SubManifold(Manifold(M)) : V; N = M-1
+    if M == 1 && V === ℝ1
+        return :(Chain{V,1}(SVector(v[1]/t[1][1])))
+    elseif M == 2 && V === ℝ2
+        return quote
+            (a,A),(b,B),(c,C) = value(t[1]),value(t[2]),value(v)
+            x1 = (c-C*(b/B))/(a-A*(b/B))
+            return Chain{V,1}(x1,(C-A*x1)/B)
+        end
+    elseif M == 3 && V === ℝ3
+        return quote
+            dv = v/∧(t)[1]; c1,c2,c3 = value(t)
+            return Chain{V,1}(
+                (c2[2]*c3[3] - c3[2]*c2[3])*dv[1] +
+                    (c3[1]*c2[3] - c2[1]*c3[3])*dv[2] +
+                    (c2[1]*c3[2] - c3[1]*c2[2])*dv[3],
+                (c3[2]*c1[3] - c1[2]*c3[3])*dv[1] +
+                    (c1[1]*c3[3] - c3[1]*c1[3])*dv[2] +
+                    (c3[1]*c1[2] - c1[1]*c3[2])*dv[3],
+                (c1[2]*c2[3] - c2[2]*c1[3])*dv[1] +
+                    (c2[1]*c1[3] - c1[1]*c2[3])*dv[2] +
+                    (c1[1]*c2[2] - c2[1]*c1[2])*dv[3])
+        end
+    end
+    N<1 && (return :(inv(t)⋅v))
+    M > ndims(V) && (return :(tt=_transpose(t,$W); tt⋅(inv(Chain{$W,1}(t)⋅tt)⋅v)))
+    x,y,xy = Grassmann.Cramer(N) # paste this into the REPL for faster eval
     mid = [:($(x[i])∧v∧$(y[end-i])) for i ∈ 1:N-1]
     out = Expr(:call,:SVector,:(v∧$(y[end])),mid...,:($(x[end])∧v))
-    return Expr(:block,:((x1,y1)=(t[1],t[end])),xy...,
-        :(Chain{V,1}(getindex.($(Expr(:call,:./,out,:(t[1]∧$(y[end])))),1))))
+    detx = :(detx = (t[1]∧$(y[end])))
+    return Expr(:block,:((x1,y1)=(t[1],t[end])),xy...,detx,
+        :(Chain{$W,1}(column($(Expr(:call,:.⋅,out,:detx))./abs2(detx)))))
 end
 
 @generated function Base.in(v::Chain{V,1},t::SVector{N,<:Chain{V,1}}) where {V,N}
@@ -366,35 +392,62 @@ end
     end
 end
 
-@generated function Base.inv(t::SVector{N,<:Chain{V,1}} where N) where V
-    N = ndims(V)-1
-    N<1 && (return :(_transpose(SVector(inv(t[1])))))
+@generated function Base.inv(t::SVector{M,<:Chain{V,1}}) where {M,V}
+    W = M≠ndims(V) ? SubManifold(Manifold(M)) : V; N = M-1
+    N<1 && (return :(_transpose(SVector(inv(t[1])),$W)))
+    M > ndims(V) && (return :(tt = _transpose(t,$W); tt⋅inv(Chain{$W,1}(t)⋅tt)))
     x,y,xy = Grassmann.Cramer(N)
-    out = if iseven(N)
+    val = if iseven(N)
         Expr(:call,:SVector,y[end],[:($(y[end-i])∧$(x[i])) for i ∈ 1:N-1]...,x[end])
+    elseif M≠ndims(V)
+        Expr(:call,:SVector,y[end],[:($(iseven(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ 1:N-1]...,:(-$(x[end])))
     else
         Expr(:call,:SVector,:(-$(y[end])),[:($(isodd(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ 1:N-1]...,x[end])
     end
-    return Expr(:block,:((x1,y1)=(t[1],t[end])),xy...,:(_transpose(.⋆($(Expr(:call,:./,out,:((t[1]∧$(y[end]))[1])))))))
+    out = if M≠ndims(V)
+        :(vector.($(Expr(:call,:./,val,:((t[1]∧$(y[end])))))))
+    else
+        :(.⋆($(Expr(:call,:./,val,:((t[1]∧$(y[end]))[1])))))
+    end
+    return Expr(:block,:((x1,y1)=(t[1],t[end])),xy...,:(_transpose($out,$W)))
 end
 
-@generated function grad(T::SVector{N,<:Chain{V,1}} where N,W=V) where V
-    N = ndims(V)-1
+@generated function grad(T::SVector{M,<:Chain{V,1}}) where {M,V}
+    W = M≠ndims(V) ? SubManifold(Manifold(M)) : V; N = ndims(V)-1
+    M < ndims(V) && (return :(ct = Chain{$W,1}(T); map(↓(V),ct⋅inv(_transpose(T,$W)⋅ct))))
     x,y,xy = Grassmann.Cramer(N)
-    out = if iseven(N)
+    val = if iseven(N)
         Expr(:call,:SVector,[:($(y[end-i])∧$(x[i])) for i ∈ 1:N-1]...,x[end])
+    elseif M≠ndims(V)
+        Expr(:call,:SVector,y[end],[:($(iseven(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ 1:N-1]...,:(-$(x[end])))
     else
         Expr(:call,:SVector,[:($(isodd(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ 1:N-1]...,x[end])
     end
-    return Expr(:block,:(t=_transpose(T,W)),:((x1,y1)=(t[1],t[end])),xy...,:(_transpose(.⋆($(Expr(:call,:./,out,:((t[1]∧$(y[end]))[1])))),↓(V))))
+    out = if M≠ndims(V)
+        :(vector.($(Expr(:call,:./,val,:((t[1]∧$(y[end])))))))
+    else
+        :(.⋆($(Expr(:call,:./,val,:((t[1]∧$(y[end]))[1])))))
+    end
+    return Expr(:block,:(t=_transpose(T,$W)),:((x1,y1)=(t[1],t[end])),xy...,:(_transpose($out,↓(V))))
+end
+
+@generated function Base.:\(t::SVector{N,<:Chain{M,1}},v::Chain{V,1}) where {N,M,V}
+    W = M≠ndims(V) ? SubManifold(Manifold(N)) : V
+    if ndims(M) > ndims(V)
+        :(ct=Chain{$W,1}(t); ct⋅(inv(_transpose(t,$W)⋅ct)⋅v))
+    else # ndims(M) < ndims(V) ? inv(tt⋅t)⋅(tt⋅v) : tt⋅(inv(t⋅tt)⋅v)
+        :(_transpose(t,$W)\v)
+    end
+end
+function inv_approx(t::Chain{M,1,<:Chain{V,1}}) where {M,V}
+    tt = transpose(t)
+    ndims(M) < ndims(V) ? (inv(tt⋅t))⋅tt : tt⋅inv(t⋅tt)
 end
 
-Base.:\(t::Chain{V,1,<:Chain{V,1}},v::Chain{V,1}) where V = value(t)\v
-Base.:\(t::Chain{M,1,<:Chain{V,1}},v::Chain{V,1}) where {M,V} = ndims(M) < ndims(V) ? (tt=transpose(t); ((inv(tt⋅t))⋅tt)⋅v) : value(t)\v
-Base.:\(t::Chain{V,1,<:Chain{M,1}},v::Chain{V,1}) where {M,V} = ndims(M) < ndims(V) ? ((inv(t⋅transpose(t)))⋅t)⋅v : value(transpose(t))\v
+Base.:\(t::Chain{M,1,<:Chain{W,1}},v::Chain{V,1}) where {M,W,V} = value(t)\v
 Base.in(v::Chain{V,1},t::Chain{W,1,<:Chain{V,1}}) where {V,W} = v ∈ value(t)
-Base.inv(t::Chain{V,1,<:Chain{V,1}}) where V = inv(value(t))
-grad(t::Chain{V,1,<:Chain{V,1}}) where V = grad(value(t))
+Base.inv(t::Chain{V,1,<:Chain{W,1}}) where {W,V} = inv(value(t))
+grad(t::Chain{V,1,<:Chain{W,1}}) where {V,W} = grad(value(t))
 INV(m::Chain{V,1,<:Chain{V,1}}) where V = Chain{V,1,Chain{V,1}}(inv(SMatrix(m)))
 
 export vandermonde
@@ -420,8 +473,8 @@ _vandermonde(x::Chain,V) = _vandermonde(value(x),V)
 
 function vandermondeinterp(x,y,V,grid) # grid=384
     coef = vandermonde(x,y,V) # Vandermonde ((inv(X'*X))*X')*y
-    xp=[minimum(x):(maximum(x)-minimum(x))/grid:maximum(x)...]
-    yp=zeros(length(xp)).+coef[1]
+    minx,maxx = minimum(x),maximum(x)
+    xp,yp = [minx:(maxx-minx)/grid:maxx...],coef[1]*ones(grid+1)
     for d ∈ 1:length(coef)-1
         yp += coef[d+1].*xp.^d
     end # fill in polynomial terms
diff --git a/src/multivectors.jl b/src/multivectors.jl
index 93340fc..cf70ab1 100644
--- a/src/multivectors.jl
+++ b/src/multivectors.jl
@@ -36,6 +36,19 @@ Chain{V,G}(val::S) where {V,G,S<:AbstractVector{𝕂}} where 𝕂 = Chain{V,G,
     ref = SVector{bg}([:(args[$i]) for i ∈ 1:bg])
     :(Chain{V,G}($(Expr(:call,:(SVector{$bg,𝕂}),ref...))))
 end
+
+@generated function Chain{V}(args::𝕂...) where {V,𝕂}
+    bg = ndims(V); ref = SVector{bg}([:(args[$i]) for i ∈ 1:bg])
+    :(Chain{V,1}($(Expr(:call,:(SVector{$bg,𝕂}),ref...))))
+end
+
+@generated function Chain(args::𝕂...) where 𝕂
+    V = SubManifold(Manifold(length(args))); bg = ndims(V)
+    ref = SVector{bg}([:(args[$i]) for i ∈ 1:bg])
+    :(Chain{$V,1}($(Expr(:call,:(SVector{$bg,𝕂}),ref...))))
+end
+
+
 function Chain(val::𝕂,v::SubManifold{V,G}) where {V,G,𝕂}
     N = ndims(V)
     Chain{V,G}(setblade!(zeros(mvec(N,G,𝕂)),val,bits(v),Val{N}()))
@@ -76,8 +89,10 @@ Chain{V,1}(m::SMatrix{N,N}) where {V,N} = Chain{V,1}(Chain{V,1}.(getindex.(Ref(m
 Chain{V,1,Chain{W,1}}(m::SMatrix{M,N}) where {V,W,M,N} = Chain{V,1}(Chain{W,1}.(getindex.(Ref(m),:,SVector{N}(1:N))))
 
 transpose_row(t::SVector{N,<:Chain{V}},i,W=V) where {N,V} = Chain{W,1}(getindex.(t,i))
+transpose_row(t::SizedVector{N,<:Chain{V}},i,W=V) where {N,V} = Chain{W,1}(getindex.(t,i))
 transpose_row(t::Chain{V,1,<:Chain},i) where V = transpose_row(value(t),i,V)
 @generated _transpose(t::SVector{N,<:Chain{V,1}},W=V) where {N,V} = :(Chain{V,1}(transpose_row.(Ref(t),$(SVector{ndims(V)}(1:ndims(V))),W)))
+@generated _transpose(t::SizedVector{N,<:Chain{V,1}},W=V) where {N,V} = :(Chain{V,1}(transpose_row.(Ref(t),$(SVector{ndims(V)}(1:ndims(V))),W)))
 Base.transpose(t::Chain{V,1,<:Chain{V,1}}) where V = _transpose(value(t))
 Base.transpose(t::Chain{V,1,<:Chain{W,1}}) where {V,W} = _transpose(value(t),V)