# In R, we assign values to a variable with the "gets" operator (  <-  ).
# For example, here, we create a variable called x which contains the
# values 1, 0, and 1 (collected together with the "c" function).
> x <- c(1,0,1)

# If we want to see what x contains, we simply type its name:
> x
[1] 1 0 1

# Here's an easy way to create a matrix in R. First, set a variable
# equal to the desired content of each column in the matrix. I'm
# going to create a matrix with 3 rows and 4 columns, so I'll need
# four variables, which I'll call x1, x2, x3 and x4. I want x1 to
# have the values I've already given x, so I'll just make that
# assignment directly:
> x1 <- x

# Next, I assign values to x2, x3, and x4:
> x2 <- c(2,2,1)
> x3 <- c(4,3,2)
> x4 <- c(1,1,5)

# Just to confirm, here's the contents of each variable:
> x1
[1] 1 0 1
> x2
[1] 2 2 1
> x3
[1] 4 3 2
> x4
[1] 1 1 5

# Now I'm going to create a matrix which I'll call x. (This will overwrite
# my previous assignment of the variable x.) I create the matrix by "binding"
# together the columns I've created using the "cbind" function:
> x <- cbind(x1,x2,x3,x4)

# The variable x now contains the desired matrix:
> x
     x1 x2 x3 x4
[1,]  1  2  4  1
[2,]  0  2  3  1
[3,]  1  1  2  5

# Let's create a new matrix variable that we'll arbitrarily call y:
> y1 <- c(1,2,5,1)
> y2 <- c(2,3,6,1)
> y3 <- c(3,4,5,1)
> y <- cbind(y1,y2,y3)
> y
     y1 y2 y3
[1,]  1  2  3
[2,]  2  3  4
[3,]  5  6  5
[4,]  1  1  1
> 


# The matrix multiplication operator in R is %*% . So to multiply the
# two matrices we've created, we do this:
> x %*% y
     y1 y2 y3
[1,] 26 33 32
[2,] 20 25 24
[3,] 18 22 22

# I might be that we want to DO something with the results of that
# matrix multiplication, rather than just look at it. If so, we'll
# want to store it as a variable. I can name it anything without
# special characters or spaces; here, I've chosen "result" to remind
# myself that it's the RESULT of the matrix multiplication:
> result <- x%*%y
> 
> result
     y1 y2 y3
[1,] 26 33 32
[2,] 20 25 24
[3,] 18 22 22

# A student asked what R would do if we ask it to perform multiplication
# on matrices that don't conform. Here, we create an example of such
# matrices and attempt the multiplication:
> x1 <- c(1,2,2,1,2)
> x2 <- c(1,3,1,3,2)
> x3 <- c(0,1,1,0,1)
> x <- cbind(x1,x2,x3)
> x
     x1 x2 x3
[1,]  1  1  0
[2,]  2  3  1
[3,]  2  1  1
[4,]  1  3  0
[5,]  2  2  1
> y <- c(1,1,2,1,2)
> 
> x %*% y
Error in x %*% y : non-conformable arguments

# R is unable to perform the operation, and tells us why.
> 

# We can transpose a matrix using the t() function:
> x
     x1 x2 x3
[1,]  1  1  0
[2,]  2  3  1
[3,]  2  1  1
[4,]  1  3  0
[5,]  2  2  1
> t(x)
   [,1] [,2] [,3] [,4] [,5]
x1    1    2    2    1    2
x2    1    3    1    3    2
x3    0    1    1    0    1
> 

# And we can store the result if we want to. Here, I've chosen the
# name tx to remind myself that this is the transpose of x. Note that
# tx is a variable name, while t(x) applies the transpose function to
# the variable x.
> tx <- t(x)
> tx
   [,1] [,2] [,3] [,4] [,5]
x1    1    2    2    1    2
x2    1    3    1    3    2
x3    0    1    1    0    1