20F Discussion Section 8 Josh Tobin: http://www.math.ucsd.edu/~rjtobin/ Some Sample Solutions Question: If the subspace of solutions of Ax = 0 has a basis of 5 vectors, and if A is a 9 × 11 matrix, what is the dimension of the column space of A? Answer. The subspace of solutions of Ax = 0 has five basis vectors, this means that the dimension of the kernel of A is 5. The matrix has 11 columns, so the rank theorem tells us that dim Col A + dim Nul A = 11 So dim Col A + 5 = 11 ⇒ dim Col A = 6. Question: Is it possible for a non-homogeneous system of seven equations in six unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solution for every right-hand side of constants? Answer. Yes, it is possible for their to be a unique solution. For example: x1 = 1, x1 = 1, x2 = 2, x3 = 3, x4 = 4, x5 = 5, x6 = 6. For the second question, the answer is no: If there is a unique solution, there are no free variables, so every column is a pivot column (ie. there are six pivot columns). This means the rank is six, so the dimension of the column space is 6. The column space is a subspace of R7 . The dimension is 6, which is smaller than 7, so there is some vector b in R7 that is not in the column space. Then Ax = b has no solutions. Question: A, B are m × n and n × p matrices, and C = AB is a m × p matrix. the system Cx = 0 has only the trivial solution. Compute the dimension of the column space of B. Suggested answer. Cx = 0 has only the trivial solution, which means that the only vector in the kernel of C is the zero vector. This means dim ker C = 0 However, the question asks about the matrix B, not C, so let us think about what this means about the kernel of B. If x is in the kernel of B, then Bx = 0. So Cx = A(Bx) = 0. This means that x is also in the kernel of C. So the dimension of the kernel of B is less than or equal to the dimension of the kernel of C, hence dim ker B = 0 Finally, the rank theorem tells us dim ker B + dim Col B = p So 0 + dim Col B = p ⇒ dim Col B = p. 1 20F Discussion Section 8 Josh Tobin: http://www.math.ucsd.edu/~rjtobin/ Some Sample Questions: Any student who wants practice writing answers to these types of questions can submit answers to me, and I will grade them and return them with feedback. If you want me to grade them, give them to me before the end of week 9 (either in section or to my office, not to the drop-boxes). Question 1: A matrix has full rank, if its rank is as big as it can be given its dimensions. Explain why an m × n matrix with more rows than columns has full rank if and only if its columns are linearly independent. Question 2: If A is a subset of a vector space V , then the complement of A is the set of all vectors that are not in A. If U is a subspace of a vector space V , is the complement of U a subspace too? Question 3: If the columns of a matrix B are linearly independent, and A is some other matrix, are the columns of AB linearly independent? Question 4: Show that the intersection of two vector subspaces is a vector subspace. Question 5: Let A and P be square matrices of the same size, where P is invertible. Show that det(P AP −1 ) = det A. Question 6: Suppose that A is a square matrix, and det(A4 ) = 0. Can A be invertible? Explain. 2