Some Sample Solutions

20F Discussion Section 8
Josh Tobin: http://www.math.ucsd.edu/~rjtobin/
Some Sample Solutions
ˆ 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.
ˆ 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.
ˆ 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.
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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).
ˆ 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.
ˆ 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?
ˆ 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?
ˆ Question 4: Show that the intersection of two vector subspaces is a vector subspace.
ˆ 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.
ˆ Question 6: Suppose that A is a square matrix, and det(A4 ) = 0. Can A be invertible? Explain.
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