sample midterm 1 (minor 2012) - university of california, san...

Math 20F Midterm 1 Review Packet

Sample Midterm 1 (Minor 2012)


Sample Midterm 2


Other Midterm Problems


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 ]


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.


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.


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

.

sample midterm 1 (minor 2012) - university of california, san...

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