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Chapter 3: Q4P (page 147)

For given sets of basis vectors, use the Gram-Schmidt method to find an orthonormal set.

(a)A=(0,鈥2,鈥0,鈥0),鈥B=(3,鈥-4,鈥0,鈥0),鈥C=(1,鈥2,鈥3,鈥4)

(b)A=(0,鈥0,鈥0,鈥7),鈥B=(2,鈥0,鈥0,鈥5),鈥C=(3,鈥1,鈥1,鈥4)

(c)A=(6,鈥0,鈥0,鈥0),鈥B=(1,鈥0,鈥2,鈥0),鈥C=(4,鈥1,鈥9,鈥2)

Short Answer

Expert verified

(a) The orthonormal set is e1,鈥e2,鈥e3=0,鈥1,鈥0,鈥0,鈥1,鈥0,鈥0,鈥0,0,鈥0,鈥3,鈥45

(b)The orthonormal set is e1,鈥e2,鈥e3=0,鈥0,鈥0,鈥1,鈥1,鈥0,鈥0,鈥0,0,鈥1,鈥1,鈥02

(c) The orthonormal set ise1,鈥e2,鈥e3=1,鈥0,鈥0,鈥0,鈥0,鈥0,鈥1,鈥0,0,鈥1,鈥0,鈥25

Step by step solution

01

(a) Find the orthonormal set

The given set of basis vectors is

A=0,鈥2,鈥0,鈥0鈥,鈥鈥B=3,鈥-4,鈥0,鈥0鈥,鈥鈥C=1,鈥2,鈥3,鈥4鈥

First, normalize the first vector as follows:

e1=A鈫A鈫=0,鈥2,鈥0,鈥022=0,鈥2,鈥0,鈥02=0,鈥1,鈥0,鈥0

Now, projecting vector Bon the first basis and subtracting this component of vector Bas follows:

B-B路e1e1=3,鈥-4,鈥0,鈥0--40,鈥1,0,鈥0=3,鈥-4,鈥0,鈥0-0,鈥-4,0,鈥0=3,鈥0,鈥0,鈥0

Let鈥檚 normalize this vector and let it be e2.

e2=3,鈥0,鈥0,鈥032=3,鈥0,鈥0,鈥03=1,鈥0,鈥0,鈥0

Next, for vector C, project it on e1and e2, and subtracting those components from C.

C-C路e1e1-C路e2e2=1,鈥2,鈥3,鈥4-20,鈥1,0,鈥0-11,鈥0,0,0=1,鈥2,鈥3,鈥4-0,鈥2,0,鈥0-1,鈥0,0,0=0,鈥0,鈥3,鈥4

Now, normalizing the above vector, then the vector e3 is:

e3=0,鈥0,鈥3,鈥432+42=0,鈥0,鈥3,鈥49+16=0,鈥0,鈥3,鈥425=0,鈥0,鈥3,鈥45

Hence, the orthonormal set ise1,鈥e2,鈥e3=0,鈥1,鈥0,鈥0,鈥1,鈥0,鈥0,鈥0,0,鈥0,鈥3,鈥45

02

(b) Find the orthonormal set

The given set of basis vectors is

A=0,鈥0,鈥0,鈥7鈥,鈥鈥B=2,鈥0,鈥0,鈥5鈥,鈥鈥C=3,鈥1,鈥1,鈥4鈥

First, normalize the first vector as follows:

e1=A鈫A鈫=0,鈥0,鈥0,鈥772=0,鈥0,鈥0,鈥77=0,鈥0,鈥0,鈥1

Now, projecting vector Bon the first basis and subtracting this component of vector B.

B-B路e1e1=2,鈥0,鈥0,鈥5-50,鈥0,0,鈥1=2,鈥0,鈥0,鈥5-0,鈥0,0,鈥5=2,鈥0,鈥0,鈥0

Le鈥 normalize this vector and let it be e2.

e2=2,鈥0,鈥0,鈥022=2,鈥0,鈥0,鈥02=1,鈥0,鈥0,鈥0

Next, for vector C, project it on e1and e2, and subtracting those components from Cas follows:

C-C路e1e1-C路e2e2=3,鈥1,鈥1,鈥4-40,鈥0,0,鈥1-31,鈥0,0,0=3,鈥1,鈥1,鈥4-0,鈥0,0,鈥4-3,鈥0,0,0=0,鈥1,鈥1,鈥0

Now, normalizing the above vector, then the vector e3 is:

e3=0,鈥1,鈥1,鈥012+12=0,鈥1,鈥1,鈥01+1=0,鈥1,鈥1,鈥02

Hence, the orthonormal set ise1,鈥e2,鈥e3=0,鈥0,鈥0,鈥1,鈥1,鈥0,鈥0,鈥0,0,鈥1,鈥1,鈥02

03

(c) Find the orthonormal set

The given set of basis vectors is

A=6,鈥0,鈥0,鈥0鈥,鈥鈥B=1,鈥0,鈥2,鈥0鈥,鈥鈥C=4,鈥1,鈥9,鈥2鈥

First, normalize the first vector as follows:

e1=A鈫A鈫=6,鈥0,鈥0,鈥062=6,鈥0,鈥0,鈥06=1,鈥0,鈥0,鈥0

Now, projecting vector Bon the first basis and subtracting this component of vector B.

B-B路e1e1=1,鈥0,鈥2,鈥0-11,鈥0,0,鈥0=1,鈥0,鈥2,鈥0-1,鈥0,0,鈥0=0,鈥0,鈥2,鈥0

Let鈥檚 normalize this vector and let it be e2.

e2=0,鈥0,鈥2,鈥022=0,鈥0,鈥2,鈥02=0,鈥0,鈥1,鈥0

Next, for vector C, project it on e1and e2, and subtracting those components from Cgives:

C-C路e1e1-C路e2e2=4,鈥1,鈥9,鈥2-41,鈥0,0,鈥0-90,鈥0,1,0=4,鈥1,鈥9,鈥2-4,鈥0,0,鈥0-0,鈥0,9,0=0,鈥1,鈥0,鈥2

Now, normalizing the above vector, the vector e3is:

e3=0,鈥1,鈥0,鈥212+22=0,鈥1,鈥0,鈥21+4=0,鈥1,鈥0,鈥25

Hence, the orthonormal set is e1,鈥e2,鈥e3=1,鈥0,鈥0,鈥0,鈥0,鈥0,鈥1,鈥0,0,鈥1,鈥0,鈥25

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