Question

Suppose A is an mxn matrix of real numbers, suppose
**x** is in the null space of A and suppose
**y** is in the column space of A^{T}. prove
that **x** is orthogonal to **y**

Answer #1

Suppose A and B are
nonempty sets of real numbers, and that for every x
∈ A, and every y ∈ B, we have x < y. Prove that A ≤
inf(B).

Give an example of a matrix, A, whose column space is in
R3 and whose null space is in R6.
For your matrix above, is it true or false that Col A =
R3? Why or Why not?

Suppose S and T are nonempty sets of real numbers such that for
each x ∈ s and y ∈ T we have x<y.
a) Prove that sup S and int T exist
b) Let M = sup S and N= inf T. Prove that M<=N

Prove that |cos x - cosy| <
|x-y| for any x, y in the real numbers.

1) Prove that for all real numbers x and y, if x < y, then x
< (x+y)/2 < y
2) Let a, b ∈ R. Prove that:
a) (Triangle inequality) |a + b| ≤ |a| + |b| (HINT: Use Exercise
2.1.12b and
Proposition 2.1.12, or a proof by cases.)

Prove the following:
For any positive real numbers x and y, x+y ≥
√(xy)

Prove the following:
If x and y are real numbers and
x+y>20, then x>10 or y>10

5. Suppose A is an n × n matrix, whose entries are all real
numbers, that has n distinct real eigenvalues. Explain why R n has
a basis consisting of eigenvectors of A. Hint: use the “eigenspaces
are independent” lemma stated in class.
6. Unlike the previous problem, let A be a 2 × 2 matrix, whose
entries are all real numbers, with only 1 eigenvalue λ. (Note: λ
must be real, but don’t worry about why this is true)....

Let x and y be real numbers. Then prove that sqrt(x^2) = abs(x)
and abs(xy) = abs(x) * abs(y)

Let
x = {x} and y ={y} represent bounded sequences of real numbers, z =
x + y, prove the following: supX + supY = supZ where sup represents
the supremum of each sequence.

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