1. How to write a row vector? For example: \(\vec{a}=(1,2,3)\)

#library(matlib)
a<- matrix(c(1,2,3))
a
##      [,1]
## [1,]    1
## [2,]    2
## [3,]    3

2. How to write a column vector in R? For example \(\vec{c}=[9,7,6]\)

c<- c(9,7,6)
c
## [1] 9 7 6

3. Can you transpose of \(\vec{a}=(1,2,3)\) ? What happens if you take the transpose twice?

t(a) 
##      [,1] [,2] [,3]
## [1,]    1    2    3
2*t(a)
##      [,1] [,2] [,3]
## [1,]    2    4    6

4. Find the followings: A. Constant multiple of a vector: \(7\vec{a}=(1,2,3)\)

7*a
##      [,1]
## [1,]    7
## [2,]   14
## [3,]   21

B. Addition of two vectors \(\vec{a}=(1,2,3)\) , \(\vec{b}=(4,5,6)\)

a<-c(1,2,3)
b<-c(4,5,6)
a+b
## [1] 5 7 9

5. Create a sytem of linear equations in R: A. Create the coefficient matrix A and the column vector b. B.Try the command in R:

library(matlib)
## Warning: package 'matlib' was built under R version 3.5.3
A1 <- matrix(c(1, 2, -1, 2), 2, 2)
b1 <- c(2,1)
showEqn(A1, b1)
## 1*x1 - 1*x2  =  2 
## 2*x1 + 2*x2  =  1

6. Can you find the rank of the matrix?

c(R(A1), R(cbind(A1,b1)))
## [1] 2 2

7. Is the system of linear equations consistent?

all.equal(R(A1), R(cbind(A1,b1)))
## [1] TRUE

8. Can you graph the system of linear equation? Use the plotEqn() command.

plotEqn(A1,b1) 
##   x1 - 1*x2  =  2 
## 2*x1 + 2*x2  =  1

9. Solve the system on linear equation and print the answers in fraction format.

Solve(A1, b1, fractions = TRUE)
## x1    =   5/4 
##   x2  =  -3/4

10. Solve the sytem of linear equations with three unknowns.

A2 <- matrix(c(2, 1, -1,
             3, 1, -2,
             2,  -1, -2), 3, 3, byrow=TRUE)
colnames(A2) <- paste0('x', 1:3)
b2 <- c(8, 11, 3)
showEqn(A2, b2)
## 2*x1 + 1*x2 - 1*x3  =   8 
## 3*x1 + 1*x2 - 2*x3  =  11 
## 2*x1 - 1*x2 - 2*x3  =   3

11. Solve the system of linear equation. Check the answers from https://www.wolframalpha.com or solve by yourself. Are you getting the same answer from R too?

solve(A2, b2)
## x1 x2 x3 
##  2  3 -1
solve(A2) %*% b2
##    [,1]
## x1    2
## x2    3
## x3   -1

12. Plot the system of linear equations with three unknowns in 3D. Use the command plotEqn3d().

plotEqn3d(A2,b2, xlim=c(0,4), ylim=c(0,4))

13. Consider the same 3x3 matrix A1. Find the determinant of that matrix. If det(A1) does not equal 0 then find the inverse of A1.

det(A2) %*% b2 
##      [,1] [,2] [,3]
## [1,]   -8  -11   -3
inv(A2) %*% b2 
##      [,1]
## [1,]    2
## [2,]    3
## [3,]   -1

14.

A2*inv(A2)
##      x1 x2 x3
## [1,]  8 -3 -1
## [2,] -6  2  2
## [3,] 10  4 -2
inv(A2)*A2
##              
## [1,]  8 -3 -1
## [2,] -6  2  2
## [3,] 10  4 -2