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Chapter 3: Q5E (page 131)

Give a geometrical description of all subspaces of鈩�3. Justify your answer.

Short Answer

Expert verified

The subspaces of 鈩�3are the planes containing the origin, a line through the origin, and the origin itself.

Step by step solution

01

Consider the set.

A subset W of the vector space 鈩�nis called a (linear) subspace of 鈩�nif it has the following three properties:

a. W contains the zero vector in 鈩�n.

b. W is closed under addition: If w鈫�1and w鈫�2are both in W, then so is w鈫�1+w鈫�2.

c. W is closed under scalar multiplication: If w鈫�is in W and k is an arbitrary scalar, then kw鈫�is in W.

A subspace of 鈩�3is either 鈩�3itself, a plane containing the origin, a line through the origin, and the origin itself. This is because a basis of 鈩�3 can contain either 3,2,1 or 0 linearly independent vectors, respectively.

02

Final answer.

The subspaces of 鈩�3 are the planes containing the origin, a line through the origin, and the origin itself.

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Most popular questions from this chapter

Question: Consider three linearly independent vectorsv1鈫�,v2鈫�,v3鈫�in 鈩�3. Are the vectorsv1鈫�,v1鈫�+v2鈫�,v1鈫�+v2鈫�+v3鈫�linearly independent as well? How can you tell?

In Exercise 40 through 43, consider the problem of fitting a conic through mgiven pointsP1(x1,y1),.......,Pm(xm,ym) in the plane; see Exercise 53 through 62 in section 1.2. Recall that a conic is a curve in鈩�2 that can be described by an equation of the formf(x,y)=c1+c2x+c3y+c4x2+c5xy+c6y2=0 , where at least one of the coefficients is non zero.

41. How many conics can you fit through four distinct pointsP1(x1,y1),.......,P4(x4,y4)?

Show that there is a nontrivial relation among the vectors v鈫�1,v鈫�2,.....,v鈫�mif (and only if) at least one of the vectorsis a linear combination of the other vectors

v鈫�1,v鈫�2,....v鈫�i-1,v鈫�i+1,....,v鈫�m.

Give an example of a matrixAsuch thatim(A)is spanned by the vector[15].

Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥� Then find a basis of the image of A and a basis of the kernel of A.

16.[1-20-100015000001]
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