Reference no: EM13688900
Question 1. Given Ax = b, provide solutions for the following systems of equations
![1261_Draw the set Yon a two-dimensional plane1.png](https://secure.expertsmind.com/CMSImages/1261_Draw%20the%20set%20Yon%20a%20two-dimensional%20plane1.png)
Question 2. Let ![1094_Draw the set Yon a two-dimensional plane.png](https://secure.expertsmind.com/CMSImages/1094_Draw%20the%20set%20Yon%20a%20two-dimensional%20plane.png)
a) Draw the set X on a two-dimensional plane.
b) Let
for all x ∈ X}
Draw the set Yon a two-dimensional plane. Comment on the mapping.
c)
for all x ∈ X}.
Draw the set Z on a two-dimensional plane. Comment on the mapping.
Question 3. For each of the following statements, either show that it is true or give a counterexample.
a) If AB is full rank then A and B are full rank.
b) If A and B are full rank then AB is full rank.
Question 4. About Cauchy - Schwartz Inequality:
a) Suppose a ≥ 0, c ≥ 0 and for all λ ∈ R, a + 2bλ + cλ2 ≥ 0 . Show that
b ≤ √ac.
b) Given u, w ∈ Rn, explain why (u + λw)T(U + λW) ≥ 0 for all λ ∈ R.
c) Apply a. to the quadratic resulting from the expression in b. , to get the Cauchy-Schwartz Inequality:
![1130_Draw the set Yon a two-dimensional plane4.png](https://secure.expertsmind.com/CMSImages/1130_Draw%20the%20set%20Yon%20a%20two-dimensional%20plane4.png)
Question 5. Find a basis for the null space and a basis for the range space of
![878_Draw the set Yon a two-dimensional plane5.png](https://secure.expertsmind.com/CMSImages/878_Draw%20the%20set%20Yon%20a%20two-dimensional%20plane5.png)