C21 Consider the subspace \(W\) of the vector space of \(2 \times 2\) matrices, \(M_{22}\), defined by the basis vectors:

\[ w_1 = \begin{bmatrix} 2 & 1 \\ 3 & -1 \end{bmatrix} \]

\[ w_2 = \begin{bmatrix} 4 & 0 \\ 2 & 3 \end{bmatrix} \]

\[ w_3 = \begin{bmatrix} -3 & 1 \\ 2 & 1 \end{bmatrix} \]

Is \(C = \begin{bmatrix} -3 & 3 \\ 6 & -4 \end{bmatrix}\) an element of \(W\)?

# Define the basis vectors of W
w1 <- matrix(c(2, 1, 3, -1), nrow=2)
w2 <- matrix(c(4, 0, 2, 3), nrow=2)
w3 <- matrix(c(-3, 1, 2, 1), nrow=2)

# Define the matrix C
C <- matrix(c(-3, 3, 6, -4), nrow=2)

# Combine the basis vectors into a 3D array
W <- array(c(w1, w2, w3), dim=c(2, 2, 3))

# Define and run a function to solve the system of linear equations
solve_for_C <- function(W, C) {
  W <- c(W)
  C <- c(C)
  A <- matrix(W, nrow=4, ncol=3, byrow=TRUE)
  x <- solve(t(A) %*% A, t(A) %*% C)
  return(x)
}

# Use the function to solve for C
solution <- solve_for_C(W, C)

# Check if C is an element of W
is_element <- all.equal(C, solution[1] * w1 + solution[2] * w2 + solution[3] * w3)

# Print results
if (is.logical(is_element) && is_element) {
  print(TRUE)
} else {
  print(FALSE)
}
## [1] FALSE

The answer is that C is not an element of W (using the default parameters for all.equal() to determine being considered close enough).