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Take z in W", and let u represent any element = is in wt) Take z, and z in w", and let u be any element Finish the proof that W- is a subspace of R".

Take z in W", and let u represent any element = is in wt) Take z, and z in w", and let u be any element Finish the proof that W- is a subspace of R".

Advanced Engineering Mathematics
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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Let W be a subspace of R", and let W- be the set of all vectors orthogonal to W. Show that W is a subspace of R" using the following steps. a. Take z in W+, and let u represent any element of W. Then z.u= 0. Take any scalar c and show that cz is orthogonal to u. (Since u was an arbitrary element of W, this will show that cz is in w-.) b. Take z, and z, in w, and let u be any element of W. Show that z, +z, is orthogonal to u. What can you conclude about z, +z,? Why? c. Finish the proof that W+ is a subspace of R". a. How can two vectors be shown to be orthogonal? O A. Determine if the dot product of the two vectors is zero. O B. Determine if the dot product of the two vectors is greater than zero. O C. Determine if the dot product of the two vectors is less than zero.
Transcribed Image Text:Let W be a subspace of R", and let W- be the set of all vectors orthogonal to W. Show that W is a subspace of R" using the following steps. a. Take z in W+, and let u represent any element of W. Then z.u= 0. Take any scalar c and show that cz is orthogonal to u. (Since u was an arbitrary element of W, this will show that cz is in w-.) b. Take z, and z, in w, and let u be any element of W. Show that z, +z, is orthogonal to u. What can you conclude about z, +z,? Why? c. Finish the proof that W+ is a subspace of R". a. How can two vectors be shown to be orthogonal? O A. Determine if the dot product of the two vectors is zero. O B. Determine if the dot product of the two vectors is greater than zero. O C. Determine if the dot product of the two vectors is less than zero.
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