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Chapter 3: Q63E (page 145)

Consider two subspaces V and Wof 鈩�n, where V is contained in W. In Exercise 62 we learned thatdim(V)鈮�dim(W) . Show that if dim(V)=dim(W), thenV=W .

Short Answer

Expert verified

It is proved that if dim(V)=dim(W), then V=W.

Step by step solution

01

Describe the dimension of the image

dim(颈尘鈥�A)=rank(A)For any matrix A,

02

Show that if dim(V)=dim(W), then V=W

Letdim(V)=dim(W)=n

The number of vectors in a basis of Vis called the dimension of .V

There arenvectors in the basis ofVand W.

Letv1,v2,...,vnbe a basis ofVand w1,w2,...,wnbe a basis ofW.

The vectors v1,v2,...,vnare in Wbecause Vis contained in W.

If nvectors are linearly independent, then vectors will form a basis of W.

Therefore, the vectorsv1,v2,...,vn form a basis of W.

Therefore,V=W .

If dim(V)=dim(W), thenV=W .

Hence, it is proved that ifdim(V)=dim(W) , thenV=W .

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Most popular questions from this chapter

In Exercises 25 through 30, find the matrixBof the linear transformationT(x鈫�)=Ax鈫� with respect to the basis J=(v鈫�1,鈥�..,v鈫�m).

A=[42-421-2-4-24];v鈫�1=[21-2],v鈫�2=[021],v鈫�3=[101]

Give an example of a linear transformation whose kernel is the plane x+2y+3z=0in鈩�3.

In Exercise 40 through 43, consider the problem of fitting a conic throughmgiven 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�2that can be described by an equation of the form , f(x,y)=c1+c2x+c3y+c4x2+c5xy+c6y2=0where at least one of the coefficients is non zero.

40. Explain why fitting a conic through the points P1(x1,y1),.......,Pm(xm,ym)amounts to finding the kernel of anm脳6matrixA. Give the entries of the row of A.

Note that a one-dimensional subspace of the kernel of defines a unique conic, since the equationsf(x,y)=0andkf(x,y)=0describe the same conic.

Question: Are the columns of an invertible matrix linearly independent?

If a 3 x 3 matrix A represents the projection onto a plane in 鈩�3, what is rank(A).
See all solutions

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