4H.and find out an orthonormal basis for W = Span(V₁, V2, V3, V4).Apply the Gram-Schmidt process to v₁=, V₂ =2V3=V4 =-231

Given that : and We have to find an orthonormal basis for Let us apply Gram-Schmidt process to the…

Answered: ------0-- = V3 = 2 Apply the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

4
H.
and find out an orthonormal basis for W = Span(V₁, V2, V3, V4).
Apply the Gram-Schmidt process to v₁
=
, V₂ =
2
V3
=
V4 =
-2
3
1

Expert Solution

Step 1: ''Introduction to the solution''

Given that : $v subscript 1 equals open square brackets table row 1 row 0 row 1 end table close square brackets comma v subscript 2 equals open square brackets table row cell negative 2 end cell row 0 row 2 end table close square brackets comma v subscript 3 equals open square brackets table row 1 row 1 row 2 end table close square brackets space$ and $v subscript 4 equals open square brackets table row cell negative 2 end cell row 3 row 1 end table close square brackets$

We have to find an orthonormal basis for $W equals S p a n left parenthesis v subscript 1 comma v subscript 2 comma v subscript 3 comma v subscript 4 right parenthesis.$

Let us apply Gram-Schmidt process to the given vectors.

Let $w subscript 1 equals v subscript 1 equals open square brackets table row 1 row 0 row 1 end table close square brackets$ Then, $stack w subscript 1 with overbrace on top equals fraction numerator w subscript 1 over denominator open vertical bar w subscript 1 close vertical bar end fraction equals 1 half open square brackets table row 1 row 0 row 1 end table close square brackets equals open square brackets table row cell 1 half end cell row 0 row cell 1 half end cell end table close square brackets$

$w subscript 2 equals v subscript 2 minus fraction numerator open angle brackets v subscript 2 comma w subscript 1 close angle brackets over denominator open angle brackets w subscript 1 comma w subscript 1 close angle brackets end fraction w subscript 1 space space space space space equals open square brackets table row cell negative 2 end cell row 0 row 2 end table close square brackets space space left square bracket sin c e comma space open angle brackets v subscript 2 comma w subscript 1 close angle brackets equals left parenthesis negative 2 plus 0 plus 2 right parenthesis equals 0 right square bracket$

Then, the corresponding the unit vector is $stack w subscript 2 with overbrace on top equals fraction numerator w subscript 2 over denominator open vertical bar w subscript 2 close vertical bar end fraction equals fraction numerator 1 over denominator square root of 8 end fraction open square brackets table row cell negative 2 end cell row 0 row 2 end table close square brackets equals open square brackets table row cell negative fraction numerator 1 over denominator square root of 2 end fraction end cell row 0 row cell fraction numerator 1 over denominator square root of 2 end fraction end cell end table close square brackets$