Math 171A Homework Assignment #2 2022
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Math 171A Homework Assignment #2
2022
1. (10 points) Consider the linear system Ax = b where
Find all basic solutions to this linear system.
2. (10 points) Consider the linear system Ax = b where
Find all basic solutions to this linear system.
3. (10 points) Consider the linear system Ax = b where
Find all basic solutions to this linear system.
4. (10 points) Consider the linear system Ax = b where
Determine all possible values for 
such that there are no basic solu- tions to this linear system.
5. (10 points) Consider the linear system Ax = b where
Determine the value of a such that there are in
nitely many basic solutions to this linear system.
6. (10 points) For a linear system Ax = b with A ∈ Rm
n and b ∈ Rm . If every m column vectors of A are linearly independent, show that there are at most 
m(n)
basic solutions.
7. (10 points) Let l(x) = 2x1 + 3x2 − 5 and x = 
+ 
p for
Determine all possible directions p such that l(x) < l(
) for all 
> 0.
8. (10 points) Let l(x) = ax1 + bx2 , a,b ∈ R and x = 
+ p for
Find p such that kpk2 = 1 and l(
+ p) − l(
) is minimum.
9. (10 points) Let l(x) = ax1 + bx2 + 1 and x = 
+ 
p for
Determine the value of (a,b) such that l(x) = 0 for all 
∈ R.
10. (10 points) Let K be the set
Determine the set of vectors c ∈ R2 such that cTx is bounded from below and also bounded from above on K .
11. (10 points) Let K be the set
Determine the set of vectors c such that cTx is unbounded from below but bounded from above on K .
12. (10 points) Let K be the set
Describe the set of value a ∈ R such that x1 + ax2 is bounded below on K .
2022-01-22