Question 1

Find the distance between the two vectors.&#10;

Accepted Answer

The distance between two vectors $\mathbf{u}$ and $\mathbf{v}$ in 3-dimensional space is given by the formula $\sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2 + (u_3 - v_3)^2}$. Substituting the given values, we get $\sqrt{(24 - (-4))^2 + (14 - 4)^2 + (19 - 7)^2} = \sqrt{(28)^2 + (10)^2 + (12)^2} = \sqrt{784 + 100 + 144} = \sqrt{1028} = 2\sqrt{257}$.

Question 2

Compute the dot product u &#183; v.&#10;

Accepted Answer

The dot product of two vectors $\mathbf{u} = [-15, 2]$ and $\mathbf{v} = [0, 11]$ is computed as $(-15 \times 0) + (2 \times 11) = 0 + 22 = 22$.

Question 3

Find the distance between the two vectors.&#10;

Accepted Answer

The distance between two vectors $\mathbf{u}$ and $\mathbf{v}$ in a 2D space is given by the formula $\sqrt{(v_1 - u_1)^2 + (v_2 - u_2)^2}$. Substituting the given values, we get $\sqrt{(1 - (-8))^2 + (-8 - (-1))^2} = \sqrt{9^2 + (-7)^2} = \sqrt{81 + 49} = \sqrt{130}$.

Question 4

Find the distance between the two vectors.&#10;

Accepted Answer

The distance between two vectors $\mathbf{u}$ and $\mathbf{v}$ is given by the magnitude of their difference: $\|\mathbf{v} - \mathbf{u}\|$. Here, $\mathbf{v} - \mathbf{u} = (-5, -3, -4)$. The magnitude is $\sqrt{(-5)^2 + (-3)^2 + (-4)^2} = \sqrt{25 + 9 + 16} = \sqrt{50} = 5\sqrt{2}$.

Question 5

Compute the dot product u &#183; v.&#10;

Accepted Answer

The dot product $ \mathbf{u} \cdot \mathbf{v} $ is computed as $ (1)(3) + (17)(1) = 3 + 17 = 20 $.

Question 6

Find the distance between the two vectors.&#10;

Accepted Answer

The distance between two vectors is given by the formula $|\mathbf { u } - \mathbf { v }|$. So,$$|\mathbf { u } - \mathbf { v }| = |( 14,17,22 ) - ( 2,9,2 )| = |( 12,8,20 )| = \sqrt { 12^2 + 8^2 + 20^2 } = \sqrt { 144 + 64 + 400 } = \sqrt { 608 } = 4 \sqrt { 38 }$$

Question 7

Determine whether the set of vectors is orthogonal.&#10;

Accepted Answer

The dot product of the first two vectors is 0, so they are orthogonal. The dot product of the second and third vectors is also 0, so they are orthogonal. The dot product of the first and third vectors is not 0, so they are not orthogonal. Therefore, the set of vectors is not orthogonal.

Question 8

Compute the dot product u &#183; v.&#10;

Accepted Answer

The dot product $ \mathbf{u} \cdot \mathbf{v} $ is computed as $ (5 \times 6) + (15 \times 15) = 30 + 225 = 255 $.

Question 9

Find the distance between the two vectors.&#10;

Accepted Answer

The distance between two vectors $\mathbf{u} = (u_1, u_2)$ and $\mathbf{v} = (v_1, v_2)$ in a 2-dimensional space is given by the formula $\sqrt{(v_1 - u_1)^2 + (v_2 - u_2)^2}$. Substituting the given values, we get $\sqrt{(-12 - 6)^2 + (12 - (-12))^2} = \sqrt{(-18)^2 + 24^2} = \sqrt{324 + 576} = \sqrt{900} = 30$.

Question 10

Express the vector x as a linear combination of the u&#697;s.&#10;

Accepted Answer

The answer of Express the vector x as a linear...

Question 11

Find a unit vector in the direction of the given vector.&#10;

Accepted Answer

The answer of Find a unit vector in the direction...

Question 12

Express the vector x as a linear combination of the u&#697;s.&#10;

Accepted Answer

The answer of Express the vector x as a linear...

Question 13

Compute the dot product u &#183; v.&#10;

Accepted Answer

The answer of Compute the dot product u &#183; v.&#10;...

Question 14

Compute the dot product u &#183; v.&#10;  )2

Accepted Answer

The answer of Compute the dot product u &#183; v.&#10;...

Question 15

Find a unit vector in the direction of the given vector.&#10;

Accepted Answer

The answer of Find a unit vector in the direction...

Question 16

Find the orthogonal projection of y onto u.&#10;

Accepted Answer

The answer of Find the orthogonal projection of y onto...

Question 17

Find the distance between the two vectors.&#10;

Accepted Answer

The answer of Find the distance between the two vectors.&#10;...

Question 18

Determine whether the set of vectors is orthogonal.&#10;

Accepted Answer

The answer of Determine whether the set of vectors is...

Question 19

Compute the dot product u &#183; v.&#10;

Accepted Answer

The answer of Compute the dot product u &#183; v.&#10;...

Question 20

Find the distance between the two vectors.&#10;

Accepted Answer

The answer of Find the distance between the two vectors.&#10;...

Question 21

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 22

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 23

&#10;Data points: (2, 1), (3, 2), (7, 3), (8, 1)33)

Accepted Answer

The answer of  &#10;Data points: (2, 1), (3, 2),...

Question 24

Find a least-squares solution of the inconsistent system Ax = b.&#10;

Accepted Answer

The answer of Find a least-squares solution of the inconsistent...

Question 25

Let W be the subspace spanned by the u&#697;s. Write y as the sum of a vector in W and a vector orthogonal to W.&#10;

Accepted Answer

The answer of Let W be the subspace spanned by...

Question 26

Find the orthogonal projection of y onto u.&#10;

Accepted Answer

The answer of Find the orthogonal projection of y onto...

Question 27

The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.&#10;

Accepted Answer

The answer of The given set is a basis for...

Question 28

Write the word or phrase that best completes each statement or answers the question.&#10;

Accepted Answer

The answer of Write the word or phrase that best...

Question 29

The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.&#10;

Accepted Answer

The answer of The given set is a basis for...

Question 30

Compute the length of the given vector.&#10;

Accepted Answer

The answer of Compute the length of the given vector.&#10;...

Question 31

Find a QR factorization of the matrix A.&#10;

Accepted Answer

The answer of Find a QR factorization of the matrix...

Question 32

11eb3ac5_a8b0_0c92_9b48_6738f0f5f5e3_TB7504_0011eb3ac5_a8b0_0c93_9b48_9319cc613a9c_TB7504_00

Accepted Answer

The answer of 11eb3ac5_a8b0_0c92_9b48_6738f0f5f5e3_TB7504_0011eb3ac5_a8b0_0c93_9b48_9319cc613a9c_TB7504_00...

Question 33

Find a least-squares solution of the inconsistent system Ax = b.&#10;

Accepted Answer

The answer of Find a least-squares solution of the inconsistent...

Question 34

Compute the length of the given vector.&#10;

Accepted Answer

The answer of Compute the length of the given vector.&#10;...

Question 35

Solve&#160;the&#160;problem.&#10;Data points: (5, -3), (2, 2), (4, 3), (5, 1)

Accepted Answer

The answer of Solve&#160;the&#160;problem.&#10;Data points: (5, -3), (2, 2), (4,...

Question 36

Let W be the subspace spanned by the u&#697;s. Write y as the sum of a vector in W and a vector orthogonal to W.&#10;

Accepted Answer

The answer of Let W be the subspace spanned by...

Question 37

Find a QR factorization of the matrix A.&#10;

Accepted Answer

The answer of Find a QR factorization of the matrix...

Question 38

Given A and b, determine the least-squares error in the least-squares solution of Ax = b.&#10;

Accepted Answer

The answer of Given A and b, determine the least-squares...

Question 39

Write the word or phrase that best completes each statement or answers the question.&#10;

Accepted Answer

The answer of Write the word or phrase that best...

Question 40

11eb3ac5_a8b0_0c92_9b48_6738f0f5f5e3_TB7504_0011eb3ac5_a8b0_32a4_9b48_2f7140086bda_TB7504_00

Accepted Answer

The answer of 11eb3ac5_a8b0_0c92_9b48_6738f0f5f5e3_TB7504_0011eb3ac5_a8b0_32a4_9b48_2f7140086bda_TB7504_00...

Question 41

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 42

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 43

&#10;Given: The data points (-2, 2), (-1, 5), (0, 5), (1, 3), (2, 5). 44) Suppose the errors in measuring the y-values of the last two data points are greater than for the&#10;Other points. Weight these data points twice as much as the rest of the data.

Accepted Answer

The answer of  &#10;Given: The data points (-2, 2),...

Question 44

&#10;Given: The data points (-3, 2), (-2, 5), (0, 5), (2, 3), (3, 3). Suppose the errors in measuring the y-values of the last two data points are greater than for the&#10;Other points. Weight these data points half as much as the rest of the data.

Accepted Answer

The answer of  &#10;Given: The data points (-3, 2),...

Deck 6: Orthogonality and Least Squares