# Linear Algebra - Unit vectors e\_i

**URL:** <https://discourse.julialang.org/t/linear-algebra-unit-vectors-e-i/10409>\
**Category:** Specific Domains\
**Created:** [April 18, 2018, 10:14am UTC](https://discourse.julialang.org/t/linear-algebra-unit-vectors-e-i/10409 "2018-04-18T10:14:44Z")\
**Posts on this page:** 1\
**Showing post:** 15

<div class="post-metadata">

**Author:** ![Elrod](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/elrod/32/22461_2.png) [@Elrod](https://discourse.julialang.org/u/Elrod)\
**Post date:** [April 18, 2018, 3:20pm UTC](https://discourse.julialang.org/t/linear-algebra-unit-vectors-e-i/10409/15 "2018-04-18T15:20:54Z")

</div>

```julia
struct lazy_e{T} <: AbstractVector{T}
    i::Int
    n::Int
end
lazy_e(i::Int, n::Int, ::Type{T} = Int) where T = lazy_e{T}(i, n)
function Base.getindex(e::lazy_e{T}, i) where T
    @boundscheck @assert i <= e.n
    ifelse(i == e.i, one(T), zero(T))
end
Base.size(e::lazy_e) = (e.n,)

```

Then on Julia 0.7:

```julia
julia> fe(x) = (@inbounds d = x' * lazy_e(3,4); d)
fe (generic function with 1 method)

julia> using StaticArrays, BenchmarkTools

julia> x = @SVector randn(4)
4-element SArray{Tuple{4},Float64,1,4}:
 -0.02571662495110571 
 -0.2620502046882458  
 -0.8456889511406026  
  0.017055806672449558

julia> y = randn(4)
4-element Array{Float64,1}:
 -1.858893446204762 
  1.883085443252701 
 -0.7957485906904918
 -0.5190800039346426

julia> @btime fe($x)
  1.520 ns (0 allocations: 0 bytes)
-0.8456889511406026

julia> @btime fe($y)
  8.301 ns (0 allocations: 0 bytes)
-0.7957485906904918

julia> lazy_e(2,5)
5-element lazy_e{Int64}:
 0
 1
 0
 0
 0

julia> lazy_e(2,5)'
1×5 LinearAlgebra.Adjoint{Int64,lazy_e{Int64}}:
 0 1 0 0 0

```

EDIT:  
Ideally, you’d do this with a for loop and `@eval`, and perhaps some other helper macros, but:

```julia
Base.:*(A::Matrix, e::lazy_e) = A[:, e.i]

Base.:*(A::Adjoint{T,Matrix{T}}, e::lazy_e) where T = A[:, e.i]
Base.:*(A::Transpose{T,Matrix{T}}, e::lazy_e) where T = A[:, e.i]
Base.:*(A::Adjoint{T,Vector{T}}, e::lazy_e) where T = A[e.i]
Base.:*(A::Transpose{T,Vector{T}}, e::lazy_e) where T = A[e.i]

Base.:*(A::Adjoint{T,lazy_e}, e::Matrix{T}) where T = A[e.i, :]
Base.:*(A::Transpose{T,lazy_e}, e::Matrix{T}) where T = A[e.i, :]
Base.:*(A::Adjoint{T,lazy_e}, e::Vector{T}) where T = A[e.i]
Base.:*(A::Transpose{T,lazy_e}, e::Vector{T}) where T = A[e.i]

```

I had to get really specific for dispatch to actually favor the newly defined multiplication methods, and I’m not sure why. This yielded:

```julia
julia> z = randn(3,4)
3×4 Array{Float64,2}:
  0.478915 -0.542889 0.142261 -0.594415
  0.509348 0.63491 1.77353 -0.545189
 -0.659932 0.995295 1.706 -0.291947

julia> @btime $z * lazy_e(2,4)
  26.288 ns (1 allocation: 112 bytes)
3-element Array{Float64,1}:
 -0.5428894820340465
  0.6349099814045577
  0.9952951252882498

julia> @btime $z[:,2]
  26.266 ns (1 allocation: 112 bytes)
3-element Array{Float64,1}:
 -0.5428894820340465
  0.6349099814045577
  0.9952951252882498

```

The matrix multiplication functions could be edited to return views if you.

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