Prove that the union of two subspaces of 𝑉 is a subspace of 𝑉 if and only if one of the subspaces is contained in the other.

Prove that in a finite-dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list of vectors.

Suppose 𝑉 is finite-dimensional and 𝑇 ∈ ℒ(𝑉, 𝑊). Show that range 𝑇 is finite-dimensional and 𝑑𝑖𝑚 𝑉 = 𝑑𝑖𝑚 𝑛𝑢𝑙𝑙 𝑇 + 𝑑𝑖𝑚 𝑟𝑎𝑛𝑔𝑒 𝑇.

Use Cramer’s rule to solve:

Solve the following differential equations