Math 20F Midterm 1 Review Packet
Sample Midterm 1 (Minor 2012)
Math 20F Midterm 1 Review Packet
Sample Midterm 2
Math 20F Midterm 1 Review Packet
Other Midterm Problems
Math 20F Midterm 1 Review Packet
Math 20F Midterm 1 Review Packet
Problems 1 – 5: Prove if the statement is true. If the statement is false, either prove it or give a counter example. For each statement that is false, what condition would you add to make it true?1. ( A+B)2=A2+2 AB+B2
2. If AB=B , then A=I .3. If A
2=0 , then A=0 .4. If AB=AC and A≠0 , then B=C .5. If A and B are square such that AB=0 , then A=0 or B=0 .
6. For (a) and (b) below, is A invertible?a) A2=0b) A2+ A=I
Review: (Harder) Computational problems
Problems 1 – 2 (ignore the 5x5 matrix on the right hand side of the augmented bar)
Let A=[a1 … a7 ] .[ A I 5×5 ]
=[0 1 −1 −1 0 −1 −20 −2 1 −1 0 −3 20 −3 −1 −9 −1 −23 −30 5 2 16 1 36 50 12 5 39 1 79 11
I 5×5 ]~ [0 1 0 2 0 4 00 0 1 3 0 5 20 0 0 0 1 6 10 0 0 0 0 0 00 0 0 0 0 0 0
−1 −1 0 0 0−2 −1 0 0 05 4 −1 0 04 3 1 1 0
17 13 1 0 1]
1. Express a4 , a6 , and a7 as a linear combination of the other column vectors.
2. Does the system Ax=bhave a solution for every b? Give a proof or a counterexample.
3. Given the equations below, determine if A is invertible.
a)
A[ 145 ]=[−1
02 ]
,
A[032 ]=[ 405 ]
,
A[004 ]=[ 8010]
b)
A[ 145 ]=[−1
02 ]
,
A[032 ]=[ 405 ]
,
A[004 ]=[353 ]
Math 20F Midterm 1 Review Packet
4. Let A be the following matrix.
A=[ 0 2 1 00 1 0 0β 3 2 31 5 1 1 ]
a) For what values of is A invertible?b) Assuming A is singular (not invertible), find all solutions to Ax=0 .
5. Let A and A−1 be the following matrices. For each B, find B
−1by looking at the relationship
between B and A.
A=[ 2 1 0−4 −1 −33 1 2 ]
,
A−1=[ 1 −2 −3−1 4 6−1 1 2 ]
a)
B=[−4 −1 −32 1 03 1 2 ]
b)
B=[ 2 1 0−4 −1 −330 10 20 ]
c)
B=[ 2 1 0−4 −1 −323 11 2 ]
6. Let A be a 4×4 matrix of all ones.
A=[1 1 1 11 1 1 11 1 1 11 1 1 1 ]
a) Show A2=4 A .
b) Let B=A+2 I . Show 8 B−B2−12 I=0c) Find B
−1.
7. Let A be the below matrix.
Math 20F Midterm 1 Review Packet
A=[1 1 1 10 1 0 02 0 2
3 01 −1 1 −1
]~ [1 1 1 10 1 0 00 0 − 4
3 −20 0 0 −2
]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.
p(1)=b1 , p' (0 )=b2 , ∫−1
1p ( x )dx=b3 , and p(−1)=b4
Review: Conceptual questions
1. Given the system, Ax=b , indicate if each statement is true or false and explain.a) If there is a unique solution, the columns of A are independent.b) If there is more than 1 solution, then the columns of A are dependent.c) If there are no solutions, then the columns of A do not span R
m.
d) If the columns of A are independent, then there is a unique solution.e) If the columns of A are dependent, then there are infinitely many solutions.f) If the columns of A do not span R
m, then there are no solutions.
g) If the columns of A span Rm
, then there is a unique solution.
2. Select the right choice and explain.i) The system must have a nontrivial solution.ii) The system cannot have a nontrivial solution.iii) Both choice i) and choice ii) are possible depending on A.
a) Suppose Ax=0 is a system of 3 linear homogeneous equations in 5 variables.b) Suppose Ax=0 is a system of 5 linear homogeneous equations in 3 variables.
3. 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 Ax=b is not consistent, then the # of pivot columns < mb) If Ax=b is consistent and n<m , then there are infinitely many solutions.c) If Ax=b is consistent and n<m , then there is exactly one solution.d) If Ax=b is consistent and the # of pivot columns = m, then there is exactly one solution.e) If Ax=b is consistent and the # of pivot columns < n, then there are infinitely many
solutions.f) If the # of pivot columns = n, then Ax=b is consistent for every b.
4. A is an m by n matrix has r pivot columns. What is the relationship between m, n, and r in each case?
a) A has an inverse.b) Ax=b has a unique solution for every b in Rm.
Math 20F Midterm 1 Review Packet
c) Ax=b has a unique solution for some, but not all b in Rm.d) Ax=b has infinitely many solution for every b in Rm.
Review: Proofs
1. Ax=b1 and Ax=b2 are both consistent systems. Is Ax=b1+b2 consistent and why?
2. Let A=span{u , v } and let B=span {u , v , u+v }. Prove A=B .
3. A and B are square matrices. If AB is invertible, prove the following:a) B is invertible.b) A is invertible.
4. Suppose S={v1 ,…, vn} is a linearly independent in Rn
and it spans Rn
, and A is an n×n
invertible matrix.
a) Prove the following set B={Av1 ,…, Avk} is independent when k≤n .
b) Prove the following set C={Av1 ,… , Avn}spans Rn
.