Deck 4: Vector Spaces

Question 1

Find a matrix A such that W = Col A.

Answer 1

D

Question 2

If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space.

Answer 2

B

Question 3

Solve the problem.

Answer 3

A

Question 4

Find a matrix A such that W = Col A.

Question 5

Find a matrix A such that W = Col A.

Question 6

If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space.

Question 7

Determine whether the vector u belongs to the null space of the matrix A.

Question 8

Solve the problem.

Question 9

Solve the problem.
Let H be the set of all polynomials having degree at most 4 and rational coefficients. Determine whether H is a vector space. If it is not a vector space, determine which of the following properties
It fails to satisfy.
A: Contains zero vector
B: Closed under vector addition
C: Closed under multiplication by scalars

A)H is not a vector space; not closed under multiplication by scalars
B)H is not a vector space; does not contain zero vector
C)H is a vector space.
D)H is not a vector space; not closed under vector addition

Question 10

Determine whether the vector u belongs to the null space of the matrix A.

Question 11

Solve the problem.

Question 12

Determine which of the sets of vectors is linearly independent.

Question 13

Find an explicit description of the null space of matrix A by listing vectors that span the null space.

Question 14

Determine if the vector u is in the column space of matrix A and whether it is in the null space of A.

Question 15

Determine if the vector u is in the column space of matrix A and whether it is in the null space of A.

Question 16

Find an explicit description of the null space of matrix A by listing vectors that span the null space.

Question 17

Determine which of the sets of vectors is linearly independent.

Question 18

Find a matrix A such that W = Col A.

Question 19

Solve the problem.

Question 20

Solve the problem.

Question 21

Find a basis for the column space of the matrix.

Question 22

Assume that the matrix A is row equivalent to B. Find a basis for the row space of the matrix A.

Question 23

Find the specified change-of-coordinates matrix.

Question 24

Determine whether the signals are linearly independent.

Question 25

Determine whether the signals are linearly independent.

Question 26

Solve the problem.

Question 27

Determine which of the sets of vectors is linearly independent.

Question 28

Determine whether {v1, v2, v3} is a basis for

Question 29

Find a basis for the column space of the matrix.

Question 30

Find the new coordinate vector for the vector x after performing the specified change of basis.

Question 31

Determine whether {v1, v2, v3} is a basis for

Question 32

Find a basis for the set of all solutions to the difference equation.

Question 33

Determine which of the sets of vectors is linearly independent.

Question 34

Solve the problem.

Question 35

Determine which of the sets of vectors is linearly independent.

Question 36

Find the new coordinate vector for the vector x after performing the specified change of basis.

Question 37

Solve the problem.

Question 38

Solve the problem.
A mathematician has found 5 solutions to a homogeneous system of 40 equations in 42
variables. The 5 solutions are linearly independent and all other solutions can be
constructed by adding together appropriate multiples of these 5 solutions. Will the system
necessarily have a solution for every possible choice of constants on the right side of the
equation? Explain.

Question 39

Find the specified change-of-coordinates matrix.

Question 40

Solve the problem.
Suppose a nonhomogeneous system of 15 linear equations in 17 unknowns has a solution
for all possible constants on the right side of the equation. Is it possible to find 3 nonzero
solutions of the associated homogeneous system that are linearly independent? Explain.

Question 41

Find the general solution of the difference equation.

Question 42

Find a basis for the set of all solutions to the difference equation.

Question 43

Find the steady-state probability vector for the stochastic matrix P.

Question 44

Find the steady-state probability vector for the stochastic matrix P.
Suppose that demographic studies show that each year about 8% of a cityʹs population moves to the suburbs (and 92% stays in the city), while 4% of the suburban population moves to the city
(and 96% remains in the suburbs). In the year 2000, 64.1% of the population of the region lived in
The city and 35.9% lived in the suburbs. What percentage of the population of the region would
Eventually live in the city if the migration probabilities were to remain constant over many years?
For simplicity, ignore other influences on the population such as births, deaths, and migration into
And out of the region.

A)66.7%
B)50%
C)41.7%
D)33.3%

Question 45

Find the steady-state probability vector for the stochastic matrix P.

Question 46

Find the general solution of the difference equation.

Question 47

Solve the problem.
Suppose that demographic studies show that each year about 6% of a cityʹs population moves to the suburbs (and 94% stays in the city), while 4% of the suburban population moves to the city
(and 96% remains in the suburbs). In the year 2000, 64% of the population of the region lived in the
City and 36% lived in the suburbs. What is the distribution of the population in 2002? For
Simplicity, ignore other influences on the population such as births, deaths, and migration into and
Out of the city/suburban region.

A)61.6% in the city and 38.4% in the suburbs
B)60.5% in the city and 39.5% in the suburbs
C)59.4% in the city and 40.6% in the suburbs
D)58.3% in the city and 41.7% in the suburbs