C16 :

Calculation using R Code

T_matrix <- matrix(c(3, 2, 1, 1, 1, -3, 2, 3, 1), nrow = 4, byrow = TRUE)
## Warning in matrix(c(3, 2, 1, 1, 1, -3, 2, 3, 1), nrow = 4, byrow = TRUE): data
## length [9] is not a sub-multiple or multiple of the number of rows [4]
cat("$$ T = ", format(T_matrix, justify = "c"), " $$")
## $$ T =   3  1  2  3  2  1  3  2  1 -3  1  1  $$

Calculating manually:

The standard basis vectors in C3 are i=[1,0,0]i=[1,0,0], j=[0,1,0]j=[0,1,0], and k=[0,0,1]k=[0,0,1].

Calculate T(i), T(j), and T(k) and arrange them into a matrix:[T(i),T(j),T(k)]

Use the transformation TT provided: i=[1,0,0],

j=[0,1,0]

and k=[0,0,1].

[T(i),T(j),T(k)]

T([1,0,0])=[3,1,1,2] T([0,1,0])=[2,1,−3,3] T([0,0,1])=[1,1,0,1]

Now, we can arrange these vectors into a matrix:

T = [ 3 2 1 1 1 1 1 − 3 0 2 3 1]

C 26. Define the linear transformation:

T:P2→C2, T (a+bx+cx2) = [2a-b, b+c]

To verify whether the given function \(T : P_2 \rightarrow \mathbb{C}^2\) is a linear transformation, we need to check two conditions:

  1. Additivity: \(T(u + v) = T(u) + T(v)\) for any vectors \(u, v\) in \(P_2\).
  2. Homogeneity: \(T(cu) = cT(u)\) for any scalar \(c\) and vector \(u\) in \(P_2\).

For the given transformation \(T(a + bx + cx^2) = [2a - b, b + c]\), let’s check these conditions.

Let \(u = a_1 + b_1x + c_1x^2\) and \(v = a_2 + b_2x + c_2x^2\) be arbitrary vectors in \(P_2\), and \(c\) be an arbitrary scalar.

  1. Additivity: \[ T(u + v) = T((a_1 + b_1x + c_1x^2) + (a_2 + b_2x + c_2x^2)) \] \[ T(u + v) = T((a_1 + a_2) + (b_1 + b_2)x + (c_1 + c_2)x^2) \] \[ T(u + v) = [2(a_1 + a_2) - (b_1 + b_2), (b_1 + b_2) + (c_1 + c_2)] \]

On the other hand: \[ T(u) + T(v) = [2a_1 - b_1, b_1 + c_1] + [2a_2 - b_2, b_2 + c_2] \] \[ T(u) + T(v) = [2(a_1 + a_2) - (b_1 + b_2), (b_1 + b_2) + (c_1 + c_2)] \]

Additivity holds.

  1. Homogeneity: \[ T(cu) = T(c(a + bx + cx^2)) \] \[ T(cu) = T(ca + cbx + ccx^2) \] \[ T(cu) = [2ca - cb, cb + cc] \]

On the other hand: \[ cT(u) = c[2a - b, b + c] = [2ca - cb, cb + cc] \]

Homogeneity holds.

Since both additivity and homogeneity hold, the given function \(T\) is a linear transformation from \(P_2\) to \(\mathbb{C}^2\).

# Define the transformation function T
T <- function(coefficients) {
  a <- coefficients[1]
  b <- coefficients[2]
  c <- coefficients[3]
  
  result <- c(2*a - b, b + c)
  return(result)
}

# Verify additivity
u <- c(1, 2, 3)  # Example vector in P2
v <- c(4, 5, 6)  # Another example vector in P2

additivity_lhs <- T(u + v)
additivity_rhs <- T(u) + T(v)

# Check if additivity holds
additivity_check <- all(additivity_lhs == additivity_rhs)
cat("Additivity Check:", additivity_check, "\n")
## Additivity Check: TRUE
# Verify homogeneity
c_value <- 2  # Example scalar
cu <- c_value * u

homogeneity_lhs <- T(cu)
homogeneity_rhs <- c_value * T(u)

# Check if homogeneity holds
homogeneity_check <- all(homogeneity_lhs == homogeneity_rhs)
cat("Homogeneity Check:", homogeneity_check, "\n")
## Homogeneity Check: TRUE