If and are nonzero subspaces of , then for every vector u in .

Question

If  and  are nonzero subspaces of  , then  for every vector u in  .

Quizplus · Accepted Answer

Since both 11edaba5_20a1_edc6_a975_bbdae2f9cc0e_TBMC1039_00 and 11edaba5_20a1_edc7_a975_f72d2f2451aa_TBMC1039_00 are nonzero subspaces of 11edaba5_20a1_edc8_a975_33d05a431915_TBMC1039_00, they both contain the zero vector. Therefore, for any vector u in 11edaba5_20a1_edca_a975_35699b9a02db_TBMC1039_00, the linear combination of u and the zero vector, which is zero, will also be in both subspaces. Hence, their intersection is nonzero and contains the zero vector, which means that the entire space 11edaba5_20a1_edc9_a975_7b0ce8886788_TBMC1039_00 is contained in their sum. Therefore, for any vector u in 11edaba5_20a1_edca_a975_35699b9a02db_TBMC1039_00, it will also be in 11edaba5_20a1_edc9_a975_7b0ce8886788_TBMC1039_00.

If and Are Nonzero Subspaces of