DATA_605_Discussion_4_C20_p443_Thonn

** DATA_605_Discussion_4_C20_p443_Thonn **

#install.packages("pracma")

** Problem C20 pg443 **

Referring to example MOLT, compute S(w) two different ways.

First use the definition of S, then compute the matrix product \(C_w\) (Definition MVP)

S: \[S(x) = C^3 \rightarrow C^4, S(\left[\begin{array}{rrr} x1 \\ x2 \\ x3 \end{array} \right]) = \left[\begin{array}{rrr} 3x1 - 2x2 + 5x3 \\ x1 + x2 + x3 \\ 9x1 -2x2 + 5x3 \\ 4x2 \end{array} \right]\]

Let \[w = \left[\begin{array}{rrr} -3 \\ 1 \\ 4 \end{array} \right]\]

\[S_w\]

Let \[C = \left[\begin{array}{rrr} 3 & -2 & 5\\ 1 & 1 & 1 \\ 9 & -2 & 5 \\ 0 & 4 & 0\end{array} \right]\]

Method-1: Calculate S_w_1 directly from substuting w in S(x)

Let \[S_w1 = \left[\begin{array}{rrr} 3*-3 -2*1 +5*4\\ -3 + 1 +4 \\ 9*-3 -2*1 + 5*4 \\ 4*1\end{array} \right] = \left[\begin{array}{rrr} \\ 9 \\ 2 \\ -9 \\4 \end{array} \right] \]

Method-2: compute the matrix product \(C_w\) (Definition MVP)

C <- matrix(data = c(3,-2,5, 1,1,1, 9,-2,5, 0,4,0), nrow = 4, ncol = 3, byrow = TRUE)
C

##      [,1] [,2] [,3]
## [1,]    3   -2    5
## [2,]    1    1    1
## [3,]    9   -2    5
## [4,]    0    4    0

w <- matrix(data = c(-3,1,4),nrow=3,ncol=1, byrow=TRUE)
w

##      [,1]
## [1,]   -3
## [2,]    1
## [3,]    4

S_w_2 <- C%*%w
S_w_2

##      [,1]
## [1,]    9
## [2,]    2
## [3,]   -9
## [4,]    4

#Conclusion:
#S-Method-1 equals S-Method-1

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