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Linear Algebra With Applications
Found in: Page 224
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

8. [5422],[367-2]

The orthonormal vectors of the sequence [5422],[367-2]is [5/74/72/72/7],-2/72/75/7-4/7.

See the step by step solution

Step by Step Solution

Step 1: Determine the Gram-Schmidt process

Consider a basis of a subspace V of Rn for j=2,....,m we resolve the vector v1 into its components parallel and perpendicular to the span of the preceding vectors v1,....,vj-1,



Step 2: Apply the Gram-Schmidt process

Let the given vectors are v1=[5422],v2=[367-2].

Obtain the values of , according to the Gram-Schmidt process.


Find u1.

u1=152+42+22+225422 =175422

Step 3: Find u⃗2

Now, here is need to find out the values of v2-(u1.v2)u1 and ||v2-(u1.v2)u1|| to obtain the value of u2.

Consider the equations below.

role="math" localid="1659441637408" u1·v1=7u1·v1u1=5423v1-u1·v1u1=367-2-5422 =-225-4


v1-u1·v1u1=4+4+25+16 =49 =7

Now find role="math" localid="1659441716297" u2.


Thus, the values of u1,u2 are [5/74/72/72/7],-2/72/75/7-4/7.

Hence, the orthonormal vectors are [5/74/72/72/7],-2/72/75/7-4/7.

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