Sample Midterm 1 (Minor 2012)
Sample Midterm 2 (2010)
Sample Midterm 3
Other Midterm Problems
Review: Computational problems
Problems 1 – 5
Let .
- Find the dimension of each of the following spaces.
a)
b)
c)
- Verify the following relationship numerically.
a)
- Find a basis for each of the following spaces.
a)
b)
c)
- Express , , and as a linear combination of the other column vectors.
- Does the system have a solution for every b? Give a proof or a counterexample.
- How can you tell that quickly?
a)
b)
c)
- Given and , find .
a)
b)
c)
d)
e)
f)
g)
- Given the equations below, find .
a),,
b),,
- Let A be the following matrix.
a)For what values of is A invertible?
b)Assuming A is singular, find the , , and the .
- Let A be a matrix of all ones.
a)Show .
b)Let . Show
c)Find .
- Let A be the below matrix.
a)Verify that A is invertible.
b)Suppose b1, b2, b3, and b4 are real numbers. Show that there is exactly one P3 polynomial such that the following equations are true.
, , , and
Review: Conceptual questions
- A is an m by n matrix. Justify your answer with a proof if the statement is true, or give a counter-example or proof if the statement is false.
a)If is not consistent, then .
b)If is consistentand , then there are infinitely many solutions.
c)If is consistentand , then there is exactly one solution.
d)If is consistentand , then there is exactly one solution.
e)If is consistentand, then there are infinitely many solutions.
f)If , then is consistent for every b.
- A is an m by n matrix of rank r. What is the relationship between m, n, and r in each case?
a)A has an inverse.
b)has a unique solution for every b in Rm.
c)has a unique solution for some, but not all b in Rm.
d)has infinitely many solution for every b in Rm.
Review: Proofs
- Suppose is a basis of , and A is an invertible matrix.
a)Prove the following set is independent when .
b)Prove the following set is a basis for.
- Let and let . Prove .
- Suppose Ais .If , prove .
- Suppose . Prove A is not invertible.