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Chapter 3: Q10E (page 164)

The column vectors of a 5脳4 matrix must be linearly dependent.

Short Answer

Expert verified

The above statement is false.

If A is a matrix of order5脳4 , then the column vectors of A need not be linearly dependent.

Step by step solution

01

Concept of linearly dependence and linearly independence matrix

If A is a matrix of order m脳n, then

RankA猢�minm,n

The numbers of unknown variables in an m脳nmatrix is n.

If the rank of a matrix A is equal to the number of unknown variables of the given matrix A then the matrix A is linearly independent.

Otherwise, linearly dependent.

02

To find whether the given matrix is linearly dependent or linearly independent

Here, we have a matrix A of order5脳4 .

Then,

RankA猢�min5,4

鈬�RankA猢�4

The number of unknown variables in a5脳4 matrix is 4.

鈬�RankA=Thenumberofunknownvariables

鈬�ThematrixAislinearlyindependent.

Thus, the given matrix A is linearly independent.

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Most popular questions from this chapter

In Exercises 21 through 25, find the reduced row-echelon form of the given matrix A. Then find a basis of the image of A and a basis of the kernel of A.

23.[102401-3-134-680-131]

Give an example of a linear transformation whose kernel is the line spanned by [-112]in鈩�3

In the accompanying figure, sketch the vectorx鈫�with [x鈫�]貜=[-12], where 貜is the basis of 鈩�2consisting of the vectorsv鈫�,w鈫�.

Explain why fitting a cubic through the mpoints P1(x1,y1),......,Pm(xm,ym)amounts to finding the kernel of an mx10matrix A. Give the entries of theof row A.

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

x2+y2+z2=1

inR3.
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