discussion2

T60 Suppose that W is a vector space with dimension 5, and U and V are subspaces of W, each of dimension 3. Prove that U ∩ V contains a nonzero vector. State a more general result.

dim_W <- 5
dim_U <- 3
dim_V <- 3

U <- matrix(0, nrow = dim_W, ncol = dim_U)
V <- matrix(0, nrow = dim_W, ncol = dim_V)

U[] <- runif(dim_W * dim_U)
V[] <- runif(dim_W * dim_V)

intersection <- U %*% t(V)

# Check if there is a nonzero vector in the intersection
nonzero_vector <- sum(intersection) != 0

if (nonzero_vector) {
  print("There exists a nonzero vector in the intersection of U and V.")
} else {
  print("There is no nonzero vector in the intersection of U and V.")
}

## [1] "There exists a nonzero vector in the intersection of U and V."