Linear algebra with applications 2nd edition by holt test bank

Linear Algebra with Applications 2nd edition by Holt Test Bank Link full download solution manual: https://findtestbanks.com/download/linear-algebra-with-applications-2nd-edition-by-ho

Linear Algebra with Applications 2nd edition by

Holt Test Bank

Link full download solution manual:

https://findtestbanks.com/download/linear-algebra-with-applications-2nd-edition-by-holt-solution-manual/

Link full download test bank: https://findtestbanks.com/download/linear-algebra-with-applications-2nd-edition-by-holt-test-bank/

Chapter 2

Euclidean Space

1 Determine where

u =

, v =

, and w =

5

Ans: 1

0

3

x

1

+ 2x = 4

−2x + 5x

= 1

Ans:

2x − x + 5x

= −1

5x + 7x

= 1

= 6

4 Express the given system of linear equations as a single vector equation

−x + x

= 3

1 2 36

x

5 Express2x+thex given−2x system=1 of linear equations as a single vector equation

−x + x + x = 1

7x

+ 3x

−

x

= 1

Ans:

=

x

7

vectors x= 3 − s

x = s

1

x

x

= 3 − s

+ 3s

vectors

x = 3 + s

Ans: x2

0

x

4

0

9 Fi nd a = 1, b = −1

a

the unknowns in the given vector equation

a

Ans : a = 2, b = −1, c = 0

10 Express b as a linear combination of the other vectors, if possible

a

1

4

, a 2 1 14

=

= , b =

2

3a − 2a = b

Ans:

11 Express b as a linear combination of the other vectors, if possible

a = 1 , a 2 = 2 , a 3 = 1 , b = 3

−2a + 2a + a = b

1

Ans : Fals e

True or False: If u and v are vectors, and c and d are scalars, then

,

v

, and

w

are vectors, then

u − (v + w) = ( u − v) + (u − w)

14 True or False: If , then

graph of u + v

2

1

Ans:

16 Determine how to divide a total mass of 18 kg among the vectors

9

u 1 , u

3 2 so that the center of mass is 2

9

1

2

9

Ans: Place 10 kg at u1 , 1 kg at u2 , and 7 kg at u3

17 Find an example of a linear system with two equations and three variables that has

s as the general solution

x

2

x

x

+ x

− 5x

= 5

Ans: A possible answer is

x − x − x

= −1

2.2 Span

u1 = 2 , u2 1 =

2

u1 = 4 , u 2 =

6

0

3 Determine if b is in the span of the other given vectors If so, write b as a linear combination

of the other vectors

a1 = , a 2 =

, b =

Ans: b = a

− a

4 Determine if b is in the span of the other given vectors If so, write b as a linear combination

of the other vectors

=

1

Ans: b is not in the span of a1 and a2

x

+ x

5 Find A , x , and b such that Ax = b corresponds to the given linear system

−x

Ans: A = 1 1

1 2

2

x1

1 , x = x

2 , and b=

4

3 8

such that Ax = b corresponds to the given linear system

+ 4x

= 3

0 x1 1

−x + 5x 1= 1 1 , and b= Ans: A= 0 2 4 , x= x 3

2

1 05 x3 1

7 Express the given system of linear equations as a vector equation 2x − x + x = 1

1 x +2x 2 + 4x 0 − x = 0 1 1

Ans: x1 x2 x 3 x4

1 2

4 10

8 Determine if the columns of the given matrix span R2 4 2

1 0

Ans: Yes, the columns span R2

9 Determine if the columns of the given matrix span R3 1 1

1 2

1 3

Ans: No, the columns do not span R3

10 Determine if the system (where x and b have the appropriate number of components) has a solution for all choices of

Ax = b b

A = 1 2

2 1

Ans: Yes, a solution exists

11 Find all values of h such that the vectors span R2

a1 = h , a2 = 2

2 h

Ans: All real numbers, except

12 given vectors span R 3 ? For what value(s) of h do the h ≠ ±2

1 4 7

2 , h , 8

3 6 9

Ans: All real numbers, except

13 True or False : Suppose a matrix A has n rows and m columns, with Then the

h ≠ 5

m < n

columns of A do not span Rn

m > n

14 True or False: Suppose a matrix A has n rows and m columns, with Then the

columns of A span Rn

Ax = b

15

rows and m columns do not span Rn ,

True or False: If the columns of a matrix A withn

then there exists a vector in Rn such that does not have a solution

m ≥

16

n

with n rows and m columns spans Rn , then

True or False: If the columns of a matrix

Ans: True

2.3 Linear Independence

2 , v =

2

Ans: Linearly independent

u = 2 , v = 0 , w = 4

Ans: Not linearly independent

1 0 2

u =

0

, v = 2 , w = 4

3

Ans: Linearly independent

3 2 2 3

Ans: Linearly independent

2 1 2

32 2

Ans: Linearly independent

1 2 3

8

7

9

7 Determine if the homogeneous system has any nontrivial solutions, where

31

A = 2 2 \

8 A x = 0

Ans:

has only the trivial solution

Determine if the homogeneous system has any nontrivial solutions, where

1 2 3

A =

1

0

1

Ax = 0

9 Determine by inspection (that is, with only minimal calculations) if the given vectors form

a linearly dependent or linearly independent set Justify your answer

u =

4 , v = 20 , w = 4

Ans: Linearly dependent, by Theorem 2.14

10 Determine if one of the given vectors is in the span of the other vectors

u =

11 True or Fal se: w = −u + v. A has n Suppose matrix rows and m columns, with Then the columns

of A are linearly dependent n < m

12 True or False: Suppose a matrix A has n rows and m columns, with Then the

columns of A are linearly independent n ≥ m

Ans: False

13 True or False: Suppose there exists a vector x such that Ax = b Then the columns of A are

linearly independent

14 True or False: If

{u , u }

for every , then the columns of A are linearly independent

is linearly {u , u } {u , u } {u , u u }

15 True or False: If , , and are all linearly independent, then

,

independent

Ans: False