- Let be vectors in
ℝn. The parallelepiped determined by these vectors
P is defined as
Now let A be an n × n matrix. Show that
is also a parallelepiped.
- In the context of Problem 1, draw P where
e1,e2 are the standard basis vectors for
ℝ2. Thus e1 =
. Now suppose
where E is the elementary matrix which takes the third row and adds to the first.
In other words, draw the result of doing E to the vectors in P. Next draw the results
of doing the other elementary matrices to
- In the context of Problem 1, either draw or describe the result of doing elementary matrices
to P. Describe geometrically the conclusion of Corollary
- Consider a permutation of . This is an ordered list of numbers taken from this
list with no repeats,
. Define the permutation matrix
P as the
matrix which is obtained from the identity matrix by placing the
jth column of I as the ijth
column of P
. Thus the 1 in the ijth column of this permutation matrix
occurs in the jth slot. What does this permutation matrix do to the column vector
- ↑Consider the 3 × 3 permutation matrices. List all of them and then determine the
dimension of their span. Recall that you can consider an m × n matrix as something in
- Determine which matrices are in row reduced echelon form.
- Row reduce the following matrices to obtain the row reduced echelon form. List the pivot
columns in the original matrix.
- Find the rank and nullity of the following matrices. If the rank is r, identify r columns in
the original matrix which have the property that every other column may be written as a
linear combination of these.
- Find the rank of the following matrices. If the rank is r, identify r columns in the original
matrix which have the property that every other column may be written as a
linear combination of these. Also find a basis for the row and column spaces of the
- Suppose A is an m × n matrix. Explain why the rank of A is always no larger than
- Suppose A is an m × n matrix in which m ≤ n. Suppose also that the rank of A equals m.
Show that A maps Fn onto Fm. Hint: The vectors e1,
,em occur as columns in the row
reduced echelon form for A.
- Suppose A is an m×n matrix and that m > n. Show there exists b ∈ Fm such that there is
no solution to the equation
- Suppose A is an m × n matrix in which m ≥ n. Suppose also that the rank of A equals n.
Show that A is one to one. Hint: If not, there exists a vector, x≠0 such that Ax = 0, and
this implies at least one column of A is a linear combination of the others. Show this would
require the column rank to be less than n.
- Explain why an n × n matrix A is both one to one and onto if and only if its rank is
- Suppose A is an m×n matrix and is a linearly independent set of vectors in
⊆ Fm. Suppose also that Azi = wi. Show that is also linearly
- Show rank
≤ rank +
- Suppose A is an m×n matrix, m ≥ n and the columns of A are independent. Suppose also
that is a linearly independent set of vectors in
Fn. Show that is
- Suppose A is an m × n matrix and B is an n × p matrix. Show that
Hint: Consider the subspace, B
∩ ker and suppose a basis for this subspace is
. Now suppose is a basis for ker
. Let be such
Bzi = wi and argue that
- Let m < n and let A be an m × n matrix. Show that A is not one to one.
- Let A be an m × n real matrix and let b ∈ ℝm. Show there exists a solution, x to the
Next show that if x,x1 are two solutions, then Ax = Ax1. Hint: First show
T = ATA. Next show if x ∈ ker
, then Ax = 0. Finally apply
the Fredholm alternative. Show ATb ∈ ker(ATA)⊥. This will give existence of a
- Show that in the context of Problem 20 that if x is the solution there, then
≤ for every
y. Thus Ax is the point of A which is closest
b of every point in A. This is a solution to the least squares problem.
- ↑Here is a point in ℝ4 :
T. Find the point in span which is
closest to the given point.
- ↑Here is a point in ℝ4 :
T. Find the point on the plane described by
x + 2y − 4z + 4w = 0 which is closest to the given point.
- Suppose A,B are two invertible n × n matrices. Show there exists a sequence of row
operations which when done to A yield B. Hint: Recall that every invertible matrix is a
product of elementary matrices.
- If A is invertible and n×n and B is n×p, show that AB has the same null space as B and
also the same rank as B.
- Here are two matrices in row reduced echelon form
Does there exist a sequence of row operations which when done to A will yield B?
- Is it true that an upper triagular matrix has rank equal to the number of nonzero entries
down the main diagonal?
- Let be vectors in
Fn. Describe a systematic way to obtain a vector vn which
is perpendicular to each of these vectors. Hint: You might consider something like
where vij is the jth entry of the vector vi. This is a lot like the cross product.
- Let A be an m×n matrix. Then ker is a subspace of
Fn. Is it true that every subspace
of Fn is the kernel or null space of some matrix? Prove or disprove.
- Let A be an n×n matrix and let Pij be the permutation matrix which switches the ith and
jth rows of the identity. Show that PijAPij produces a matrix which is similar to A which
switches the ith and jth entries on the main diagonal.
- Recall the procedure for finding the inverse of a matrix on Page 133. It was shown that the
procedure, when it works, finds the inverse of the matrix. Show that whenever the matrix
has an inverse, the procedure works.
- If EA = B where E is invertible, show that A and B have the same linear relationships
among their columns.