R, 117 144 137 133 129129 123 bytes
Reasonably verbose at the moment. Should be able to shave a few more out. Gained a number of bytes formatting it correctly, but saved some swapping out the matrix for an array.
Thanks to Alex for the tip on the replacing sep with s and removing the function name.
Dropped the array completely and used a series of reps to build each line.
Although soundly beaten by Miff, his solution made me realise I could drop the s=' ' altogether.
function(i){s=sum(i);i=cumsum(i);b=0;for(n in 1:s){d=i[i>=n][1];cat(c(rep(0,b),rep(1,d-b),rep(0,s-d)),fill=T,s=' ');if(d==n)b=d}}
And the test
> (function(i){s=sum(i);i=cumsum(i);b=0;for(n in 1:s){d=i[i>=n][1];cat(c(rep(0,b),rep(1,d-b),rep(0,s-d)),fill=T,s=' ');if(d==n)b=d}})(c(5,1,1,3,1))
1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1 1 1 0
0 0 0 0 0 0 0 1 1 1 0
0 0 0 0 0 0 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0 1
>