diff --git a/src/Unitful.jl b/src/Unitful.jl
index 343f49b6..0b9a42ce 100644
--- a/src/Unitful.jl
+++ b/src/Unitful.jl
@@ -21,8 +21,8 @@ import Base: steprange_last, unsigned
 end
 
 import Dates
-import LinearAlgebra: Diagonal, Bidiagonal, Tridiagonal, SymTridiagonal
-import LinearAlgebra: istril, istriu, norm
+import LinearAlgebra: Diagonal, Bidiagonal, Tridiagonal, SymTridiagonal, Adjoint, Transpose, AdjOrTransAbsMat
+import LinearAlgebra: istril, istriu, norm, mul!, dot, /, \, inv, pinv
 import Random
 
 import ConstructionBase: constructorof
@@ -69,5 +69,6 @@ include("logarithm.jl")
 include("complex.jl")
 include("pkgdefaults.jl")
 include("dates.jl")
+include("linearalgebra.jl")
 
 end
diff --git a/src/linearalgebra.jl b/src/linearalgebra.jl
new file mode 100644
index 00000000..77433605
--- /dev/null
+++ b/src/linearalgebra.jl
@@ -0,0 +1,84 @@
+
+# Multiplication
+
+function mul!(C::StridedVecOrMat{<:AbstractQuantity{T}}, 
+              A::StridedMatrix{<:AbstractQuantity{T}}, 
+              B::StridedVecOrMat{<:AbstractQuantity{T}},
+              alpha::Number, beta::Number) where {T<:Base.HWNumber}
+    _mul!(C, A, B, alpha, beta)
+end
+
+function mul!(C::StridedVecOrMat{<:AbstractQuantity{T}}, 
+              A::AdjOrTransAbsMat{<:AbstractQuantity{T}, <:StridedMatrix}, 
+              B::StridedVecOrMat{<:AbstractQuantity{T}},
+              alpha::Number, beta::Number) where {T<:Base.HWNumber}
+    _mul!(C, A, B, alpha, beta)
+end
+
+function _mul!(C, A, B, alpha, beta)
+    if unit(beta) != NoUnits
+        throw(DimensionError("beta", 1.0))
+    elseif unit(eltype(C)) != unit(eltype(A)) * unit(eltype(B)) * unit(alpha)
+        throw(DimensionError("A * B .* α", "C"))
+    end
+    C0 = ustrip(C)
+    A0 = ustrip(A)
+    B0 = ustrip(B)
+    mul!(C0, A0, B0)
+    _linearalgebra_count()
+    return C
+end
+
+function dot(A::StridedArray{<:AbstractQuantity{T}}, 
+             B::StridedArray{<:AbstractQuantity{T}}) where {T<:Base.HWNumber}
+    A0 = ustrip(A)
+    B0 = ustrip(B)
+    C0 = dot(A0, B0)
+    _linearalgebra_count()
+    C = C0 * unit(eltype(A)) * unit(eltype(B))
+    return C
+end
+
+# Division
+
+function (\)(A::StridedMatrix{<:AbstractQuantity{T}}, 
+             B::StridedVecOrMat{<:AbstractQuantity{T}}) where {T<:Base.HWNumber}
+    A0 = ustrip(A)
+    B0 = ustrip(B)
+    C0 = A0 \ B0
+    _linearalgebra_count()
+    u = unit(eltype(B)) / unit(eltype(A))
+    Tu = typeof(one(eltype(C0)) * u)
+    return reinterpret(Tu, C0)
+end
+
+function (/)(A::StridedVecOrMat{<:AbstractQuantity{T}}, 
+             B::StridedVecOrMat{<:AbstractQuantity{T}}) where {T<:Base.HWNumber}
+    A0 = ustrip(A)
+    B0 = ustrip(B)
+    C0 = A0 / B0
+    _linearalgebra_count()
+    u = unit(eltype(A)) / unit(eltype(B))
+    Tu = typeof(one(eltype(C0)) * u)
+    return reinterpret(Tu, C0)
+end
+
+function inv(A::StridedMatrix{<:AbstractQuantity{T}}) where {T<:Base.HWNumber}
+    C0 = inv(ustrip(A))
+    _linearalgebra_count()
+    u = inv(unit(eltype(A)))
+    Tu = typeof(one(eltype(C0)) * u)
+    return reinterpret(Tu, C0)
+end
+
+function pinv(A::StridedMatrix{<:AbstractQuantity{T}}; kw...) where {T<:Base.HWNumber}
+    C0 = pinv(ustrip(A); kw...)
+    _linearalgebra_count()
+    u = inv(unit(eltype(A)))
+    Tu = typeof(one(eltype(C0)) * u)
+    return reinterpret(Tu, C0)
+end
+
+# This function is re-defined during testing, to check we hit the fast path:
+_linearalgebra_count() = nothing
+
diff --git a/src/utils.jl b/src/utils.jl
index 53f12b95..85b1bda7 100644
--- a/src/utils.jl
+++ b/src/utils.jl
@@ -28,6 +28,27 @@ true
 @inline ustrip(u::Units, x) = ustrip(uconvert(u, x))
 @inline ustrip(T::Type, u::Units, x) = convert(T, ustrip(u, x))
 
+"""
+    ustrip(u::Units, xs::AbstractArray{<:Quantity})
+
+This broadcasts `ustrip.(u, xs)`, unless `xs isa StridedArray` whose units match `u`,
+in which case it reinterprets, which saves making a copy.
+
+```jldoctest
+julia> ustrip(u"m", [1, 2, 3]u"m") isa Base.ReinterpretArray{Int}  # fast path
+true
+
+julia> ustrip(u"m", [1, 2, 3]u"mm") == [1//1000, 2//1000, 3//1000]  # mismatch requires slow path
+true
+```
+"""
+ustrip(u::Units, xs::AbstractArray) = ustrip.(u, xs)
+function ustrip(u::Units, xs::StridedArray{T}) where {T}
+    dimension(u) == dimension(T) || return ustrip.(u, xs)
+    isequal(promote(true * u, oneunit(T))...) || return ustrip.(u, xs)
+    return reinterpret(numtype(T), xs)
+end
+
 """
     ustrip(x::Number)
     ustrip(x::Quantity)
@@ -50,7 +71,8 @@ true
 @inline ustrip(x::Missing) = missing
 
 """
-    ustrip(x::Array{Q}) where {Q <: Quantity}
+    ustrip(x::Array{Q}) where {Q <: Quantity{T}}}
+
 Strip units from an `Array` by reinterpreting to type `T`. The resulting
 `Array` is a not a copy, but rather a unit-stripped view into array `x`. Because the units
 are removed, information may be lost and this should be used with some care.
@@ -75,7 +97,7 @@ julia> a[1] = 3u"m"; b
  2
 ```
 """
-@inline ustrip(A::Array{Q}) where {Q <: Quantity} = reinterpret(numtype(Q), A)
+@inline ustrip(A::StridedArray{Q}) where {Q <: Quantity} = reinterpret(numtype(Q), A)
 
 @deprecate(ustrip(A::AbstractArray{T}) where {T<:Number}, ustrip.(A))
 
@@ -90,6 +112,15 @@ ustrip(A::Diagonal) = Diagonal(ustrip(A.diag))
 ustrip(A::Bidiagonal) = Bidiagonal(ustrip(A.dv), ustrip(A.ev), ifelse(istriu(A), :U, :L))
 ustrip(A::Tridiagonal) = Tridiagonal(ustrip(A.dl), ustrip(A.d), ustrip(A.du))
 ustrip(A::SymTridiagonal) = SymTridiagonal(ustrip(A.dv), ustrip(A.ev))
+ustrip(A::Adjoint) = adjoint(ustrip(parent(A)))
+ustrip(A::Transpose) = transpose(ustrip(parent(A)))
+
+ustrip(u::Units, A::Diagonal) = Diagonal(ustrip(u, A.diag))
+ustrip(u::Units, A::Bidiagonal) = Bidiagonal(ustrip(u, A.dv), ustrip(u, A.ev), ifelse(istriu(A), :U, :L))
+ustrip(u::Units, A::Tridiagonal) = Tridiagonal(ustrip(u, A.dl), ustrip(u, A.d), ustrip(u, A.du))
+ustrip(u::Units, A::SymTridiagonal) = SymTridiagonal(ustrip(u, A.dv), ustrip(u, A.ev))
+ustrip(u::Units, A::Adjoint) = adjoint(ustrip(u, parent(A)))
+ustrip(u::Units, A::Transpose) = transpose(ustrip(u, parent(A)))
 
 """
     unit(x::Quantity{T,D,U}) where {T,D,U}
diff --git a/test/runtests.jl b/test/runtests.jl
index aec5a66f..321e35c0 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -84,6 +84,66 @@ const colon = Base.:(:)
     @test ConstructionBase.constructorof(typeof(1.0m))(2) === 2m
 end
 
+@testset "LinearAlgebra functions" begin
+    CNT = Ref(0)
+    Unitful._linearalgebra_count() = (CNT[] += 1; nothing)
+    @testset "> Matrix multiplication: *" begin
+        M = rand(3,3) .* u"m"
+        M_ = view(M,:,1:3)
+        v = rand(3) .* u"V"
+        v_ = view(v, 1:3)
+
+        CNT[] = 0
+
+        @test unit(first(M * M)) == u"m*m"
+        @test M * M == M_ * M == M * M_ == M_ * M_
+
+        @test unit(first(M * v)) == u"m*V"
+        @test M * v == M_ * v == M * v_ == M_ * v_
+
+        VERSION >= v"1.3" && @test CNT[] == 10
+
+        @test unit(first(v' * M)) == u"m*V"
+        @test v' * M == v_' * M == v_' * M == v_' * M_
+
+        VERSION >= v"1.3" && @test CNT[] == 15
+
+        @test unit(v' * v) == u"V*V"
+        @test v' * v == v_' * v == v_' * v == v_' * v_
+
+        VERSION >= v"1.3" && @test CNT[] == 20
+
+        # Mixed with & without units
+        N = rand(3,3)
+        w = rand(3)
+
+        CNT[] = 0
+        
+        @test unit(first(M * N)) == u"m"
+        @test unit(first(N * M)) == u"m"
+
+        @test unit(first(M * w)) == u"m"
+        @test unit(first(N * v)) == u"V"
+
+        @show CNT[] # not specialised yet
+
+    end
+    @testset "> Matrix multiplication: mul!" begin
+        A = rand(3,3) .* u"m"
+        B = rand(3,3) .* u"m"
+        C = fill(zero(eltype(A*B)), 3, 3)
+        CNT[] = 0
+
+        mul!(C, A, B)
+        if VERSION >= v"1.3" # the 5-arm mul! exists
+            mul!(C, A, B, true, true)
+            mul!(C, A, B, 3, 7) # not specialised yet
+
+            @show CNT[]
+        end
+    end
+end
+
 @testset "Types" begin
     @test Base.complex(Quantity{Float64,NoDims,NoUnits}) ==
         Quantity{Complex{Float64},NoDims,NoUnits}
@@ -1251,6 +1311,16 @@ end
             @test_deprecated ustrip([1,2])
             @test ustrip.([1,2]) == [1,2]
             @test typeof(ustrip([1u"m", 2u"m"])) <: Base.ReinterpretArray{Int,1}
+
+            # With target type
+            @test @inferred(ustrip(u"m", [1, 2]u"m")) == [1,2]
+            @test @inferred(ustrip(u"km", [1, 2]u"m")) == [1//1000, 2//1000]
+            @test typeof(ustrip(u"m", [1, 2]u"m")) <: Base.ReinterpretArray{Int,1}
+            @test typeof(ustrip(u"m/ms", [1, 2]*(u"km/s"))) <: Base.ReinterpretArray{Int,1}
+
+            # Structured matrices
+            @test typeof(ustrip(adjoint([1,2]u"m"))) <: Adjoint{Int}
+            @test typeof(ustrip(transpose([1 2; 3 4]u"m"))) <: Transpose{Int}
             @test typeof(ustrip(Diagonal([1,2]u"m"))) <: Diagonal{Int}
             @test typeof(ustrip(Bidiagonal([1,2,3]u"m", [1,2]u"m", :U))) <:
                 Bidiagonal{Int}
@@ -1258,6 +1328,9 @@ end
                 Tridiagonal{Int}
             @test typeof(ustrip(SymTridiagonal([1,2,3]u"m", [4,5]u"m"))) <:
                 SymTridiagonal{Int}
+
+            @test typeof(ustrip(u"m", adjoint([1,2]u"m"))) <: Adjoint{Int}
+            @test typeof(ustrip(u"m", Diagonal([1,2]u"m"))) <: Diagonal{Int}
         end
         @testset ">> Linear algebra" begin
             @test istril([1 1; 0 1]u"m") == false