- Let be vectors in ℝ
^{n}. The parallelepiped determined by these vectors Pis defined asNow let A be an n × n matrix. Show that

is also a parallelepiped.

- In the context of Problem 1, draw Pwhere e
_{1},e_{2}are the standard basis vectors for ℝ^{2}. Thus e_{1}=,e_{2}=. Now supposewhere E is the elementary matrix which takes the third row and adds to the first. Draw

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 P. - In the context of Problem 1, either draw or describe the result of doing elementary matrices
to P. Describe geometrically the conclusion of Corollary 4.3.7.
- Consider a permutation of . This is an ordered list of numbers taken from this list with no repeats,. Define the permutation matrix Pas the matrix which is obtained from the identity matrix by placing the j
^{th}column of I as the i_{j}^{th}column of P. Thus the 1 in the i_{j}^{th}column of this permutation matrix occurs in the j^{th}slot. What does this permutation matrix do to the column vector^{T}? - ↑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
F
^{nm}. - 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
matrices.
- Suppose A is an m × n matrix. Explain why the rank of A is always no larger than
min.
- Suppose A is an m × n matrix in which m ≤ n. Suppose also that the rank of A equals m.
Show that A maps F
^{n}onto F^{m}. Hint: The vectors e_{1},,e_{m}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 ∈ F
^{m}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 n.
- Suppose A is an m×n matrix and is a linearly independent set of vectors in A⊆ F
^{m}. Suppose also that Az_{i}= w_{i}. Show thatis also linearly independent. - Show rank ≤ 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 F
^{n}. Show thatis linearly independent. - Suppose A is an m × n matrix and B is an n × p matrix. Show that
Hint: Consider the subspace, B

∩ kerand suppose a basis for this subspace is. Now supposeis a basis for ker. Letbe such that Bz_{i}= w_{i}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 ∈ ℝ
^{m}. Show there exists a solution, x to the systemNext show that if x,x

_{1}are two solutions, then Ax = Ax_{1}. Hint: First show that^{T}= A^{T}A. Next show if x ∈ ker, then Ax = 0. Finally apply the Fredholm alternative. Show A^{T}b ∈ ker(A^{T}A)^{⊥}. This will give existence of a solution. - 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 Awhich is closest to b of every point in A. This is a solution to the least squares problem.
- ↑Here is a point in ℝ
^{4}:^{T}. Find the point in spanwhich is closest to the given point. - ↑Here is a point in ℝ
^{4}:^{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? Explain.

- 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 F
^{n}. Describe a systematic way to obtain a vector v_{n}which is perpendicular to each of these vectors. Hint: You might consider something like thiswhere v

_{ij}is the j^{th}entry of the vector v_{i}. This is a lot like the cross product. - Let A be an m×n matrix. Then keris a subspace of F
^{n}. Is it true that every subspace of F^{n}is the kernel or null space of some matrix? Prove or disprove. - Let A be an n×n matrix and let P
^{ij}be the permutation matrix which switches the i^{th}and j^{th}rows of the identity. Show that P^{ij}AP^{ij}produces a matrix which is similar to A which switches the i^{th}and j^{th}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.

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