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Chapter 3: Q65E (page 145)

Consider two subspaces V and Wof鈩�n , with .role="math" localid="1660906447344" V鈭�W={0鈫�}
What is the relationship among dim(V), dim(W), anddim(V+W) ? (For the definition of V+W, see Exercise 3.2.50; Exercise 3.2.51 is helpful.) 1

Short Answer

Expert verified

The required relationship isdim(V+W)=k+l=dimV+dimW .

Step by step solution

01

Describe the dimension of the image

For any matrix A,

dim(颈尘鈥�A)=rank(A)

02

Find the relationship among dim(V), dim(W), and dim(V+W)

It is known that is BW={w1,鈥�w2,...,wk}base ofW andBV={v1,鈥�v2,...,vl} is base of Vthen vector form BW鈭�BV={w1,鈥�w2,...,wk,鈥�v1,鈥�v2,...,vl}are linearly independent.

Now, it is known that since they are independent then they form a basis fromk+l dimensional space, and it is known that they span W+V. Therefore,

dim(V+W)=k+l=dimV+dimW

Hence, the required relationship is .dim(V+W)=k+l=dimV+dimW

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Most popular questions from this chapter

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.

18.1151012201230124

Find a basis of the kernel of the matrix

[1203500146]

Justify your answer carefully; that is, explain how you know that the vectors you found are linearly independent and span the kernel.

Give an example of a parametrization of the ellipse

x2+y24=1

in鈩�2 . See Example .

Give an example of a matrixAsuch thatim(A)is the plane with normal vector [132]in鈩�3鈩�3 .

Give an example of a function whose image is the unit sphere

x2+y2+z2=1

inR3.
See all solutions

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