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Let w,=(1,0,0,1) and Let W={w | w•w, w w, denotes the dot product of w;=(-1,0,0,1). w.w2 = 0}, where = W, & w. (a) Show that W is a vector space (over the field R). (b) Find the basis of W. (c) Find the dimension of W. (d) Is the answer to part (c) unique? k

Let w,=(1,0,0,1) and Let W={w | w•w, w w, denotes the dot product of w;=(-1,0,0,1). w.w2 = 0}, where = W, & w. (a) Show that W is a vector space (over the field R). (b) Find the basis of W. (c) Find the dimension of W. (d) Is the answer to part (c) unique? k

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,=(1,0,0,1) and w,=(-1,0,0,1). Let W={w | w•w1 w w denotes the dot product of W, & w. (a) Show that W is a vector space (over the field R). (b) Find the basis of W. (c) Find the dimension of W. (d) Is the answer to part (c) unique? w.W2 w w, = 0}, where k
Transcribed Image Text:Let w,=(1,0,0,1) and w,=(-1,0,0,1). Let W={w | w•w1 w w denotes the dot product of W, & w. (a) Show that W is a vector space (over the field R). (b) Find the basis of W. (c) Find the dimension of W. (d) Is the answer to part (c) unique? w.W2 w w, = 0}, where k
Expert Solution
Step 1

Given vectors are w1→=1,0,0,1, w2→=-1,0,0,1.

Let W=w→| w→.w1→=w→.w2→=0.

Let w→=a,b,c,d∈W. Therefore:

                        w→.w1→=0a,b,c,d1,0,0,1=0a+d=0a=-d                            …1

and:

                       w→.w1→=0a,b,c,d-1,0,0,1=0-a+d=0a=d                                …2

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