mathWHIZ

Math 220 Linear Algebra (Spring 2018)

Homework 2.3 and Practice problems for Exam 1

Due Thursday March 1

These two problems from Section 2.3 and the sample midterm problems will be graded on

completeness only, so it is not necessary to start every problem on a new page. Please make

sure that both assignments ((Homework 2.3 and Practice problems for Exam 1)) must be

submitted as a PDF file that rotated correctly to be graded.

1. (4 points) Let A be a 5⇥5 matrix such that Ax = 0 has only the trivial solution. Is it possible for there to be a vector b in IR5 for which Ax = b has no solution? Justify your answer.

2. (4 points) Let T : IRn ! IRn be an invertible linear transformation. Let A be the matrix representation of T . What can you say about the row reduced echelon form of A?

Don’t forget the sample midterm problems! Please attach all of your work together

(do not submit 2.3 separately from the sample midterm problems).

Practice problems for Exam 1

Due Thursday March 1

1. Solve the following system of equations using row-reduction.

2x1 � 2x2 + 4x3 = 6 3x1 + x2 � 2x3 = 5 �x1 + 3x2 + x3 = �5

2. Solve the following systems of equations for x1, x2, x3,... where the systems of equations have the following augmented matrices. Write the solution in vector parametric form where

applicable.

(a)

2

4 1 2 �1 0 | 0 0 1 0 �2 | 1 0 0 1 �3 | 0

3

5

(b)

2

4 1 6 �3 | 2 0 1

1 2

| 3 �1 �8 2 | �1

3

5

3. Determine the value(s) of h, if any, such that the linear system of equations represented by the following augmented matrix has

2

4 1 �1 | 4 2 �1 | 10

�2 2 | h

3

5

(a) no solution,

(b) exactly one solution,

(c) infinitely many solutions

4. Given the matrices A, B, C, and D evaluate or write is not defined:

A =

2

4 1 �1 0 3

2 0

3

5 B =

2

4 1 0 �1

�2 0 0 0 1 0

3

5 C =  3 �1 9 3

� D =

 2 4

1 2

�

a) 2D b) AC c) AB d) C�1 e) D�1

f) BT g) C + D h) (C + D)T i) C�T j) A�T

1

5. Given the size of the matrices A, B, C, and D, determine whether the following operations are defined, and if defined, determine the size of the resulting matrix.

A 3⇥2 B 2⇥2 C 2⇥3 D 3⇥3

(a) 5A

(b) B � C (c) (A � CT )T

(d) CT AT + 2DT

6. Determine whether the set of vectors are linearly independent. Give reasons for your answer.

(a)

8 <

:

2

4 16

�8 12

3

5 ,

2

4 �4 2

�4

3

5

9 =

;

(b)

8 <

:

2

4 2

�5 1

3

5 ,

2

4 �6 5

3

3

5 ,

2

4 0

0

0

3

5

9 =

;

(c)

8 <

:

2

4 5

5

1

3

5 ,

2

4 �1 0

1

3

5 ,

2

4 3

6

1

3

5 ,

2

4 2

0

0

3

5

9 =

;

(d)

8 <

:

2

4 1

1

3

3

5 ,

2

4 �3 0

4

3

5 ,

2

4 5

�1 2

3

5

9 =

;

7. Given Ax = b is represented by the augmented matrix

2

4 �1 1 �3 �6 �4 �3 3 �8 �16 �10 2 �2 7 14 10

3

5 g

2

4 1 �1 3 6 4 0 0 1 2 2

0 0 0 0 0

3

5

(a) How many solutions does the system of equations have?

(b) Are the columns of A independent? Explain.

(c) Do the columns of A span IR3?

(d) Is Ax = b consistent for every b? Explain.

(e) If T : IRn ! IRm, is a linear transformation given by T(x) = Ax, what is n? What is m?

8. T : IR2 ! IR3 is a linear transformation such that T [(1, 1)] = (�1, 0, 4) and T [(1, 0)] = (5, 3, 2). Find (a) T [(2, 2)], and (b) T [(2, 1)].

9. True or False: For all n⇥n matrices A, B, and C (no explanation required):

(a) A + B = B + A

(b) (A + B)T = AT + BT

(c) (AT )T = A

(d) If AB = AC then B = C.

(e) If B = C then AB = AC.

10. Determine A�1 if A =

2

4 1 1 2

2 3 �2 3 3 7

3

5

11. True or False: For an arbitrary nxm matrix A (no explanation required):

(a) The equation 3x1 + e ⇡x2 = cos

� ⇡ 12

� x3 + 4 is a linear equation.

(b) Every matrix is row-equivalent to a unique matrix in echelon form.

(c) A system of two linear equations with two unknowns can have exactly two solutions.

(d) If the coe�cient matrix A is row equivalent to a row-echelon form matrix that has a pivot in every row, then the system of linear equations, Ax = b, is always consistent.

(e) A homogeneous system of linear equations is always consistent.

(f) The columns of an m⇥n matrix A span IRm if and only if the equation Ax = b is consistent for every vector b in IRm.

(g) If a set contains fewer vectors than there are entries in the vectors, then the set is linearly

independent.

(h) Every linear transformation rotates a vector.

(i) Not every linear transformation from IR n to IR

m is a matrix transformation.

(j) If A can be row reduced to the identity matrix, then A must be invertible.

(k) If the columns of A are linearly independent, then A is invertible.

12. A classmate claims the inverse of the following matrix A is B. Prove that the classmate is right or wrong.

A =

2

4 1 �1 �2 2 �3 �5

�1 3 5

3

5 B =

2

4 0 1 1

5 �3 �1 �3 1 0

3

5

13. Given that A, B, and C are all invertible nxn matrices, find the inverse of ABC and prove it is the inverse.