4.6. EXERCISES 127

19. Let m < n and let A be an m× n matrix. Show that A is not one to one.

20. Let A be an m× n real matrix and let b ∈ Rm. Show there exists a solution, x to thesystem

ATAx = ATb

Next show that if x,x1 are two solutions, then Ax = Ax1. Hint: First show that(ATA

)T= ATA. Next show if x ∈ ker

(ATA

), then Ax = 0. Finally apply the Fred-

holm alternative. Show ATb ∈ ker(ATA)⊥. This will give existence of a solution.

21. Show that in the context of Problem 20 that if x is the solution there, then |b−Ax| ≤|b−Ay| for every y. Thus Ax is the point of A (Rn) which is closest to b of everypoint in A (Rn). This is a solution to the least squares problem.

22. ↑Here is a point in R4 : (1, 2, 3, 4)T. Find the point in span



1

0

2

3

 ,

0

1

3

2

 which

is closest to the given point.

23. ↑Here is a point in R4 : (1, 2, 3, 4)T. Find the point on the plane described by x+2y−

4z + 4w = 0 which is closest to the given point.

24. Suppose A,B are two invertible n× n matrices. Show there exists a sequence of rowoperations which when done to A yield B. Hint: Recall that every invertible matrixis a product of elementary matrices.

25. If A is invertible and n× n and B is n× p, show that AB has the same null space asB and also the same rank as B.

26. Here are two matrices in row reduced echelon form

A =

 1 0 1

0 1 1

0 0 0

 , B =

 1 0 0

0 1 1

0 0 0

Does there exist a sequence of row operations which when done to A will yield B?Explain.

27. Is it true that an upper triagular matrix has rank equal to the number of nonzeroentries down the main diagonal?

28. Let {v1, · · · ,vn−1} 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 somethinglike this

det

e1 e2 · · · en

v11 v12 · · · v1n...

......

v(n−1)1 v(n−1)2 · · · v(n−1)n

where vij is the jth entry of the vector vi. This is a lot like the cross product.

29. Let A be an m × n matrix. Then ker (A) is a subspace of Fn. Is it true that everysubspace of Fn is the kernel or null space of some matrix? Prove or disprove.