## Description

Quiz – 2

Please Note :

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Group 1

(One among the following set of questions)

1. (3 points) Consider the problem of computing the orthonormal basis of the subspace using the Grahm Schmidt Approach. Let W denote the subspace spanned by the vectors {v1,v2,v3}โR3 where

๏ฃฎ ๏ฃน ๏ฃฎ ๏ฃน ๏ฃฎ ๏ฃน

1 0 c

๏ฃฏ ๏ฃบ ๏ฃฏ ๏ฃบ ๏ฃฏ ๏ฃบ v1 = ๏ฃฏ๏ฃฏ a ๏ฃบ๏ฃบ, v2 = ๏ฃฏ๏ฃฏ 0 ๏ฃบ๏ฃบ and, v3 = ๏ฃฏ๏ฃฏ 0 ๏ฃบ๏ฃบ

๏ฃฏ ๏ฃบ ๏ฃฏ ๏ฃบ ๏ฃฏ ๏ฃบ

๏ฃฐ ๏ฃป ๏ฃฐ ๏ฃป ๏ฃฐ ๏ฃป

0 b 0 (1)

The first entry of the third orthonormal basis vector for the subspace W is (Please Note: Apply the Grahm Schmit algorithm by considering vectors in the following order, i.e., {v1,v2,v3})

2. (3 points) Consider the problem of computing the orthonormal basis of the subspace using the Grahm Schmidt Approach. Let W denote the subspace spanned by the vectors {v1,v2,v3}โR3 where

๏ฃฎ ๏ฃน ๏ฃฎ ๏ฃน ๏ฃฎ ๏ฃน

1 0 c

๏ฃฏ ๏ฃบ ๏ฃฏ ๏ฃบ ๏ฃฏ ๏ฃบ v1 = ๏ฃฏ๏ฃฏ a ๏ฃบ๏ฃบ, v2 = ๏ฃฏ๏ฃฏ 0 ๏ฃบ๏ฃบ and, v3 = ๏ฃฏ๏ฃฏ 0 ๏ฃบ๏ฃบ

๏ฃฏ ๏ฃบ ๏ฃฏ ๏ฃบ ๏ฃฏ ๏ฃบ

๏ฃฐ ๏ฃป ๏ฃฐ ๏ฃป ๏ฃฐ ๏ฃป

0 b 0 (4)

The second entry of the third orthonormal basis vector for the subspace W is (Please Note: Apply the Grahm

Schmit algorithm by considering vectors in the following order, i.e., {v1,v2,v3})

Group 2

(One among the following set of questions)

1. (3 points) Consider the problem of expressing the vector v โ R3 as a linear combination of combination of the

๏ฃฎ ๏ฃน x

๏ฃฏ ๏ฃบ

basis vectors i.e., v . Let v = ๏ฃฏ๏ฃฏ y ๏ฃบ๏ฃบ. Compute the coefficient of linear

๏ฃฏ ๏ฃบ

๏ฃฐ ๏ฃป z

combination ฮฑ3.

2. (3 points) Consider the problem of expressing the vector v โ R3 as a linear combination of combination of the

๏ฃฎ ๏ฃน x

๏ฃฏ ๏ฃบ

basis vectorsi.e., v = P3i=1 ฮฑibi. Let v = ๏ฃฏ๏ฃฏ y ๏ฃบ๏ฃบ. Compute the coefficient of linear

๏ฃฏ ๏ฃบ

๏ฃฐ ๏ฃป z

๏ฃด๏ฃณ b1 b2 b3 ๏ฃด๏ฃพ

combination ฮฑ1.

3. (3 points) Consider the problem of expressing the vector v โ R3 as a linear combination of combination of the

๏ฃฎ ๏ฃน x

๏ฃฏ ๏ฃบ

basis vectors i.e., v . Let v = ๏ฃฏ๏ฃฏ y ๏ฃบ๏ฃบ. Compute the coefficient of linear

๏ฃฏ ๏ฃบ

๏ฃฐ ๏ฃป z

combination ฮฑ2.

Group 3

(One among the following set of questions)

1. (1 point) For some a,b โRn, that satisfy b ฬธ= ฮฑa, Projab = 0

A. True B. False

2. (1 point) For all a,b โRn, that satisfy b ฬธ= ฮฑa, Projab = 0

A. True B. False

Group 4

(One among the following set of questions)

1. (1 point) Consider all possible sets of 3 linearly independent vectors (denoted by say, a,b,c โR3) such that b โฅ a.

Let d = cโProjbcโProjac. Then, d โฅ b and d โฅ a.

A. True B. False

2. (1 point) Consider all possible sets of 3 linearly independent vectors (denoted by say, a,b,c โ R3) . Let d = cโProjbcโProjac. Then, d โฅ b and d โฅ a.

A. True B. False

Group 5

1. Consider all possible a,b,c โR3 such that a โฅ b and b โฅ c. Then a โฅ c.

A. True B. False

Group 6

(One among the following set of questions)

๏ฃฎ ๏ฃน a

๏ฃฏ ๏ฃบ

1. (3 points) The coordinate vector of x โR3 in basis. If xB = ๏ฃฏ๏ฃฏ b ๏ฃบ๏ฃบ, the

๏ฃฏ ๏ฃบ

๏ฃฐ ๏ฃป c

first entry of x in the standard basis is

๏ฃฎ ๏ฃน a

๏ฃฏ ๏ฃบ

2. (3 points) The coordinate vector of x โR3 in basis. If xB = ๏ฃฏ๏ฃฏ b ๏ฃบ๏ฃบ, the

๏ฃฏ ๏ฃบ

๏ฃฐ ๏ฃป c

second entry of x in the standard basis is

๏ฃฎ ๏ฃน a

๏ฃฏ ๏ฃบ

3. (3 points) The coordinate vector of x โR3 in basis. If xB = ๏ฃฏ๏ฃฏ b ๏ฃบ๏ฃบ, the

๏ฃฏ ๏ฃบ

๏ฃฐ ๏ฃป c

๏ฃด๏ฃณ b1 b2 b3 ๏ฃด๏ฃพ

third entry of x in the standard basis is

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