All Levels of a Factor in a Model Matrix in R

I have a data.frame consisting of numeric and factor variables as seen below.

testFrame <- data.frame(First=sample(1:10, 20, replace=T),
           Second=sample(1:20, 20, replace=T), Third=sample(1:10, 20, replace=T),
           Fourth=rep(c("Alice","Bob","Charlie","David"), 5),
           Fifth=rep(c("Edward","Frank","Georgia","Hank","Isaac"),4))

I want to build out a matrix that assigns dummy variables to the factor and leaves the numeric variables alone.

model.matrix(~ First + Second + Third + Fourth + Fifth, data=testFrame)

As expected when running lm this leaves out one level of each factor as the reference level. However, I want to build out a matrix with a dummy/indicator variable for every level of all the factors. I am building this matrix for glmnet so I am not worried about multicollinearity.

Is there a way to have model.matrix create the dummy for every level of the factor?

(Trying to redeem myself...) In response to Jared's comment on @Fabians answer about automating it, note that all you need to supply is a named list of contrast matrices. contrasts() takes a vector/factor and produces the contrasts matrix from it. For this then we can use lapply() to run contrasts() on each factor in our data set, e.g. for the testFrame example provided:

> lapply(testFrame[,4:5], contrasts, contrasts = FALSE)
$Fourth
        Alice Bob Charlie David
Alice       1   0       0     0
Bob         0   1       0     0
Charlie     0   0       1     0
David       0   0       0     1

$Fifth
        Edward Frank Georgia Hank Isaac
Edward       1     0       0    0     0
Frank        0     1       0    0     0
Georgia      0     0       1    0     0
Hank         0     0       0    1     0
Isaac        0     0       0    0     1

Which slots nicely into @fabians answer:

model.matrix(~ ., data=testFrame, 
             contrasts.arg = lapply(testFrame[,4:5], contrasts, contrasts=FALSE))

You need to reset the contrasts for the factor variables:

model.matrix(~ Fourth + Fifth, data=testFrame, 
        contrasts.arg=list(Fourth=contrasts(testFrame$Fourth, contrasts=F), 
                Fifth=contrasts(testFrame$Fifth, contrasts=F)))

or, with a little less typing and without the proper names:

model.matrix(~ Fourth + Fifth, data=testFrame, 
    contrasts.arg=list(Fourth=diag(nlevels(testFrame$Fourth)), 
            Fifth=diag(nlevels(testFrame$Fifth))))

All Levels of a Factor in a Model Matrix in R

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