Vector and Matrix Algebra, Exams of Algebra

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Vector and Matrix Algebra

Lecture notes

Packages for this section

This is (almost) all base R! We only need this for one thing later: library (tidyverse)

Adding a number to a vector

Define a vector, then “add 2” to it: u

[1] 2 3 6 5 7

k <- 2 u + k

[1] 4 5 8 7 9

adds 2 to each element of u.

Scalar multiplication

As per linear algebra: k

[1] 2

u

[1] 2 3 6 5 7

k ***** u

[1] 4 6 12 10 14

Each element of vector multiplied by 2.

Combining different-length vectors

No error here (you get a warning). What happens? u

[1] 2 3 6 5 7

w <- c (1, 2) u + w

Warning in u + w: longer object length is not a

multiple of shorter object length

[1] 3 5 7 7 8

Add 1 to first element of u, add 2 to second. Go back to beginning of w to find something to add: add 1 to 3rd element of u, 2 to 4th element, 1 to 5th.

How R does this

Keep re-using shorter vector until reach length of longer one. “Recycling”. If the longer vector’s length not a multiple of the shorter vector’s length, get a warning (probably not what you want). Same idea is used when multiplying a vector by a number: the number keeps getting recycled.

Adding matrices

What happens if you add two matrices? A

[,1] [,2]

[1,] 1 3

[2,] 2 4

B

[,1] [,2]

[1,] 5 6

[2,] 7 8

A + B

[,1] [,2]

[1,] 6 9

[2,] 9 12

Adding matrices

Nothing surprising here. This is matrix addition as we and linear algebra know it.

Multiplying matrices?

Not matrix multiplication (as per linear algebra). Elementwise multiplication. Also called Hadamard product of A and B.

Legit matrix multiplication

Like this: A

[,1] [,2]

[1,] 1 3

[2,] 2 4

B

[,1] [,2]

[1,] 5 6

[2,] 7 8

A %%* B

[,1] [,2]

[1,] 26 30

[2,] 38 44

but…

except that M is not an R matrix, and thus this doesn’t work: v <- c (1, 3) M %%* v

Error in M %*% v: requires numeric/complex matrix/vector ar

Making a genuine matrix

Do this first: M <- as.matrix (M)

and then all is good: M %%* v

[,1]

[1,] 37

[2,] 29

[3,] 21

Matrix inverse

To find the inverse of A: A

[,1] [,2]

[1,] 1 3

[2,] 2 4

solve (A)

[,1] [,2]

[1,] -2 1.

[2,] 1 -0.

You can check that the matrix inverse and equation solution are correct.

Inner product

Vectors in R are column vectors, so just do the matrix multiplication (t() is transpose): a <- c (1, 2, 3) b <- c (4, 5, 6) t (a) %%* b

[,1]

[1,] 32

Note that the answer is actually a 1 × 1 matrix. Or as the sum of the elementwise multiplication: sum (a ***** b)

[1] 32