R-Studio Notebook Vectors Matrices.Rmd

---
title: "R Notebook: Lesson 4, Vectors and Matrices"
output: html_notebook
---

Creating and Modifying Vectors 
```{r}
V <- c(1,2,3,4,5,6)
print(V[1:3])
print(V[5])
V2 <- V[3:5]
print(V2)
V3 <- 1:9
V3[10] <- 10
print(V3)

```
Basic Vector Operations

```{r}
print(V+1)
print(V*2)
print(V[1:3] / V2)
crossprod(V[1:3], V2)
```

Creating and Modifying Matrices
```{r}
A <- matrix(1:9, nrow = 3, byrow = TRUE)
B <- matrix(1:2, ncol = 2)
I <- diag(3)
print("MATRIX A")
print(A)

print("MATRIX B")
print(B)

print("Identity Matrix I")
print(I)

colnames(A) <- c("price", "store no.", "inventory")
rownames(A) <- c("item1", "item2", "item3")
print(A)
print(A[1:2,])
print(A[,2:3])
```

Linear Algebra Examples
```{r}
#Multiplying by the Identity Matrix
print(A*I)

#Matrix Multiplication of a matrix mxn by a matrix nxp (or, in this case, nxn)
E <- matrix(2:7, nrow = 2)
F <- matrix(2:7, ncol = 2)
print(E%*%F)

# Matrix Muliplied by a Vector
g <- c(7,17)
print(F%*%g)

# Identity
solve(E%*%F)

# Determinant
det(E%*%F)

# Row
row(A)

# Col
col(A)

# Boolean matrix, produced by comparison
row(A)==col(A)

# Diag
diag(A)
```

Generating a Covariance Matrix
Example from the book "The Art of Programming in R" by Norman Matloff (2011)

```{r}
makecov <- function(rho,n) {
 m <- matrix(nrow=n,ncol=n)
 m <- ifelse(row(m) == col(m),1,rho)
 return(m)
}

Z <- matrix(3:8, ncol = 2)
print(Z)

X <- makecov(0.2, 5)
print(X)

print(X[,1:2]%*%t(Z))
```

Using the apply function on a matrix
Example from the book "The Art of Programming in R" by Norman Matloff (2011)

```{r}
print(Z)
 apply(Z,2,mean)
 
f <- function(x) x/c(2,8)
Y <- apply(Z,1,f)
print(Y)
```

Beware of Unintended Dimension Reduction!

```{r}
M <- matrix(1:8, ncol = 2)
print(M)

r <- M[2,]
# R is a vector, not a 1 dimensional matrix!
print(r)

print("Attributes of M: ")
print(attributes(M))
print("Attributes of r: ")
print(attributes(r))
# There are none, because r is a vector.

# Is this what we expected?
print(M%*%r)

# Using the 'drop' argument to maintain the slice as a 1x2 matrix
m <- M[2,, drop=FALSE]
print(attributes(m))
print(M%*%t(m))
```