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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

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

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.

Explain why you need at least 鈥榤鈥� vectors to span a space of dimension 鈥榤鈥�. See Theorem 3.3.4b.

Consider a 4 x 2 matrix A and 2 x 5 matrix B.

a. What are the possible dimensions of the kernel of AB?

b. What are the possible dimensions of the image of AB?

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

See all solutions

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