“Vectors” in R doesn’t have mathematical sense. Here we have a question from a learner on Coursera:

x <- 1:3
x
# [1] 1 2 3

# Transpose it
t(x)
#      [,1] [,2] [,3]
# [1,]    1    2    3

# Transpose it again
t(t(x))
#      [,1]
# [1,]    1
# [2,]    2
# [3,]    3

# Make calculations
x * t(x)
#      [,1] [,2] [,3]
# [1,]    1    4    9

Question:

1. Is x a “mathematical” vector?
2. If not, why t(x) worked?
3. Why does x * t(x) show a weird result?

---

Indeed, “vectors” in R cannot be considered mathematically. Instead, “vectors” here simply means “a series of data without dimensions”. Since vectors don’t have dimensions, nothing like “row vector” or “column vector” would be discussed, nor would they have matrix operations.

If you look up the documentation of t function, you will know why t(x) in your example worked: “When x is a vector, it is treated as a column, i.e., the result is a 1-row matrix.” Besides, the operations in R is “vectorized operations”, which is very unique. (Please go review the lecture of “Vectorized Operations” if you are not familiar with it.) Thus, x * t(t(x)) is not a matrix operation in math and returns weird results.

That being said, you can still create a “mathematical vector” by creating “a matrix with only one row or one column”, although it will be a “matrix” in R.

R_vector <- 1:3
class(R_vector)
# "integer"
# Which means R_vector is an interger vector

col_vector <- matrix(1:3)
class(col_vector)
# "matrix"
# Note that this is NOT a vector in R language

row_vector <- t(col_vector)
# row_vector <- matrix(1:3, nrow = 1)
class(row_vector)
# "matrix"

I hope the demonstration above helps you to understand. Feel free to ask if you have further questions. :)