Deck 5: Inner Product Spaces

Question 1

Find the length of the vector

A)
B)
C)
D)
E)

Answer 1

Question 2

Find given and .

A)
B)
C)
D)
E)

Answer 2

Question 3

Find the vector v with length 2 and the same direction as the vector .

A)
B)
C)
D)
E)

Answer 3

Question 4

Find the vector v with length 2 and the same direction as the vector .

A)
B)
C)
D)
E)

Question 5

Find the distance d between u= and v= .

A)
B)
C)
D)
E)

Question 6

Find , where u= and v= .

A)
B)
C)
D)
E)

Question 7

Find ,where .

A)
B)
C)
D)
E)

Question 8

Find the angle in radians between the vectors . Round your answers to three decimal places.

A)
B)
C)
D)
E)

Question 9

Find the angle in radians between the vectors . Round your answers to three decimal places.

A)
B)
C)
D)
E)

Question 10

Determine all vectors v that are orthogonal to .

A) , where t is any real number
B) , where t is any real number
C) , where t is any real number
D) , where t is any real number
E) , where t is any real number

Question 11

Determine all vectors v that are orthogonal to .

A) where t is any real numbers
B) where s and t are any real numbers
C) where s and t are any real numbers
D) where t is any real numbers
E) where s and t are any real numbers

Question 12

Find for the inner product defined in R², where .

A) 8
B) 48
C) 6
D) 58
E) 43

Question 13

Find for the inner product defined in R², where .

A)
B)
C)
D)
E)

Question 14

Find for the standard inner product defined in R³, where .

A)
B)
C)
D)
E)

Question 15

Find for the standard inner product defined in R³, where .

A)
B)
C)
D)
E)

Question 16

Find for the inner product defined in R³, where .

A)
B)
C)
D)
E)

Question 17

Use the functions in C[-1, 1] to find for the inner product .

A)
B)
C)
D)
E)

Question 18

Use the functions in C[-1, 1] to find for the inner product .

A)
B)
C)
D)
E)

Question 19

Use the functions in C[-1, 1] to find for the inner product .

A)
B)
C)
D)
E)

Question 20

Use the inner product to find , where and .

A)
B)
C)
D)
E)

Question 21

Find the angle in radians between the vectors for the inner product . Round your answers to two decimal places.

A)
B)
C)
D)
E)

Question 22

Determine whether the set of vectors in R² is orthogonal (but not orthonormal), orthonormal, or neither.

A) orthonormal
B) neither
C) orthogonal

Question 23

Determine whether the set of vectors in R³ is orthogonal (but not orthonormal), orthonormal, or neither.

A) orthonormal
B) orthogonal
C) neither

Question 24

Determine whether the set of vectors in R³ is orthogonal (but not orthonormal), orthonormal, or neither.

A) neither
B) orthonormal
C)orthogonal

Question 25

Find the coordinates of relative to the orthonormal basis in R².

A)
B)
C)
D)
E)

Question 26

Find the coordinates of relative to the orthonormal basis in R³.

A)
B)
C)
D)
E)

Question 27

Use the Gram-Schmidt orthonormalization process to transform the basis for R³ into an orthonormal basis. Use the Euclidean inner product for R³ and use the vectors in the order in which they are shown.

A)
B)
C)
D)
E)

Question 28

Use the Gram-Schmidt orthonormalization process to transform the basis for R³ into an orthonormal basis. Use the Euclidean inner product for R³ and use the vectors in the order in which they are shown.

A)
B)
C)
D)
E)

Question 29

Use the Gram-Schmidt process to change the basis into an orthonormal basis for the subspace of R³ spanned by the vector .

A)
B)
C)
D)
E)

Question 30

Find an orthonormal basis for the solution space of the following homogeneous system of linear equations.

A)
B)
C)
D)
E)

Question 31

S is the subspace of R³ consisting of the xz-plane. Find the orthogonal complement S.

A)
B)
C)
D)
E)

Question 32

Which of the following is a description of the orthogonal complement of

A)
B)
C)
D)
E)

Question 33

Find the projection of the vector onto the subspace

A)
B)
C)
D)
E)

Question 34

Find bases for the four fundamental subspaces of the matrix .

A) N(A)-basis: , N(A^T)-basis: , R(A)-basis: , and R(A^T)-basis:
B) N(A)-basis: , N(A^T)-basis: , R(A)-basis: , and R(A^T)-basis:
C) N(A)-basis: , N(A^T)-basis: , R(A)-basis: , and R(A^T)-basis:
D) N(A)-basis: , N(A^T)-basis: , R(A)-basis: , and R(A^T)-basis:
E) N(A)-basis: , N(A^T)-basis: , R(A)-basis: , and R(A^T)-basis:

Question 35

Find the least squares solution of the system , where .

A)
B)
C)
D)
E)

Question 36

Graph the points and the least squares regression line for the data points on the same set of axes.

A)
B)
C)
D)
E) none of the choices

Question 37

Find the least squares regression line for the points .

A)
B)
C)
D)
E)

Question 38

Suppose that the following table shows the annual sales, in millions of dollars, for Advanced Auto Parts and Auto Zone for 2000 through 2007. Find the least squares quadratic model for Advanced Auto Parts. Let t represents the year, with corresponding to 2000. Round all coefficients to two decimal places.

A)
B)
C)
D)
E)

Question 39

Suppose that the following table shows the annual sales, in millions of dollars, for Advanced Auto Parts and Auto Zone for 2000 through 2007. Find the least squares quadratic model for Advanced Auto Parts and use the model to predict sales for the year 10. Let t represent the year, with corresponding to 2000. Round all coefficients to two decimal places and the answer to nearest million dollars.

A) $20,686 million
B) $34,086 million
C) $14,960 million
D)$29,686 million
E) $10,448 million

Question 40

Suppose that the table shows the world carbon dioxide emissions y in millions of metric tons during the years 1999 to 2004. Find the least squares regression line for the data. Let t represent the year, with corresponding to 1999. Round your answer to four significant figures

A)
B)
C)
D)
E)

Question 41

Find the cross product of the unit vectors .

A)
B)
C)
D)
E)

Question 42

Sketch the graph of the cross product of the unit vectors , .

A)
B)
C)
D)

Question 43

Find the cross product of the vectors , .

A)
B)
C)
D)
E)

Question 44

​Show that the cross product of the vectors , is orthogonal to both u and v.

Question 45

Find the cross product of the vectors , .

A)
B)
C)
D)
E)

Question 46

​Show that the cross product of the vectors , is orthogonal to both u and v.

Question 47

Find the area of the parallelogram that has the vectors , as adjacent sides.

A) 9
B) 27
C) 729
D) 18
E) 6

Question 48

Find the area of the parallelogram that has the vectors , as adjacent sides.

A)
B) 400
C)
D) 4
E) 800

Question 49

​Verify the points are the vertices of a parallelogram, then find its area.

Question 50

Find the triple scalar product where , , .

A)
B)
C)
D)
E)

Question 51

Find the triple scalar product where .

A)
B)
C)
D)
E)

Question 52

Find the linear least squares approximating function g for the function , .

A)
B)
C)
D)
E)

Question 53

Find the quadratic least squares approximating function g for the function . Round numerical values in your answer to four decimal places.

A)
B)
C)
D)
E)

Question 54

Find the Fourier approximation of second order for the function on the interval .

A)
B)
C)
D)
E)