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Chapter 3: Q34E (page 144)

Consider a subspace V in鈩�m that is defined by homogeneous linear equations

|a11x1+a12x2+鈥�+a1mxm=0a21x1+a22x2+鈥�+a2mxm=0鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌�鈰�鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌�鈰�鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆�夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆��鈰�鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌�鈰�鈥�an1x1+a22x2+鈥�+anmxm=0|.

What is the relationship between the dimension of Vand the quantitym-n

? State your answer as an inequality. Explain carefully.

Short Answer

Expert verified

A subspaceV of 鈩�mhas dimension at least m鈭�n.

Step by step solution

01

To mention given data and recall the theorem 3.3.7

We have the subspaceV of鈩�m defined byn homogeneous linear eqautions,

role="math" localid="1660126682574" a11x1+a12x2+鈰�+a1mxm=0a21x1+a22x2+鈰�+a2mxm=0鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌�鈰�鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆�夆�夆��鈰�鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆�夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌�鈰�鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆��鈰�an1x1+an2x2+鈰�+anmxm=0,

Therefore,V can be written as

V=ker(A), whereA is ann脳m matrixA=[aij] .

Suppose the column vector bex1x2鈰�xm .

Theorem 3.3.7, is stated as follows:

For anyn脳m matrix A, the equation,

dim(ker(A))+dim(im(A))=m.

02

To find the relationship between the dimension V and m-n 

We have the rank of Arank(A)鈮�n.

Thus, by Theorem 3.3.7, we have,

dim(V)=dim(ker(A))=m鈭�rank(A)鈮�m鈭�n

鈭�dim(V)鈮�m鈭�n

.

Hence, a subspace Vof鈩�m has dimension at least m鈭�n.

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Most popular questions from this chapter

Consider two n x m matrices A and B. What can you say about the relationship among the quantities rank(A), rank(B), rank(A+B).

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

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)?

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

Determine whether the following vectors form a basis of 鈩�4; [1111],[1-11-1],[1248],[1-24-8].
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