a. Two vectors are linearly dependent if and only if they

Chapter 1, Problem 22E

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QUESTION:

In Exercises 21 and 22 mark each statement True or False. Justify each answer on the basis of a careful reading of the text.

a. Two vectors are linearly dependent if and only if they lie on a line through the origin.

b. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.

c. If x and y are linearly independent, and if z is in Span {x, y}, then {x, y, z} is linearly dependent.

d. If a set in \(\mathbb{R}^{n}\) is linearly dependent, then the set contains more vectors than there are entries in each vector.

Questions & Answers

QUESTION:

In Exercises 21 and 22 mark each statement True or False. Justify each answer on the basis of a careful reading of the text.

a. Two vectors are linearly dependent if and only if they lie on a line through the origin.

b. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.

c. If x and y are linearly independent, and if z is in Span {x, y}, then {x, y, z} is linearly dependent.

d. If a set in \(\mathbb{R}^{n}\) is linearly dependent, then the set contains more vectors than there are entries in each vector.

ANSWER:

Solution:-

Step1

We have to find whether the given statement is true or false with reason.

Step2

a. Two vectors are linearly dependent if and only if they lie on a line through the origin.

The given statement is true.

A set of vectors is linearly dependent if at least one of the vectors is a multiple of the other.

Consider the vectors

And observe that (6,2)=2(3,1)

That is,

So, one vector is a scalar multiple of the other.

Therefore, the vectors  are linearly dependent.

From figure lie on same line.

If the two vectors lie on the same line through the origin, they must be linearly dependent.

So, it is clear that two vectors are linearly dependent if and only if they lie on a line through the origin.

Step3

b. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.

The given statement is false.

Let the 3 dimensional vector space

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