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Chapter 1: Problem 9

Let \(u\) and \(v\) be distinct vectors in a vector space \(\mathrm{V}\). Show that \(\\{u, v\\}\) is linearly dependent if and only if \(u\) or \(v\) is a multiple of the other.

Short Answer

Expert verified
In this exercise, we showed that given two distinct vectors \(u\) and \(v\) in a vector space \(V\), the set \(\{u, v\}\) is linearly dependent if and only if \(u\) or \(v\) is a multiple of the other. We used the definition of linear dependence and showed that if \(u\) or \(v\) is a multiple of the other, then \(\{u, v\}\) is linearly dependent. Next, we proved that if \(\{u, v\}\) is linearly dependent, then \(u\) or \(v\) is a multiple of the other. Thus, establishing the required equivalence.

Step by step solution

01

Define linear dependence

Linear dependence is defined as the existence of non-zero coefficients c and d such that their linear combination for the given vectors equals the zero vector. That is, cu + dv = 0, where c 鈮� 0 or d 鈮� 0.
02

Prove that if u or v is a multiple of the other, then {u, v} is linearly dependent

Suppose u is a multiple of v, i.e., u = kv for some scalar k. Then, we can write the linear combination as: 1*u - k*v = 1*(kv) - k*v = 0. Since 1 鈮� 0 and k 鈮� 0 (as u and v are distinct), this implies that {u, v} is linearly dependent. Similarly, if v is a multiple of u (v = ku), then -k*u + 1*v = -k*u + (ku) = 0. Here also, both coefficients (-k and 1) are nonzero, proving that {u, v} is linearly dependent.
03

Prove that if {u, v} is linearly dependent, then u or v is a multiple of the other

Let {u, v} be a linearly dependent set. This means there exist non-zero coefficients c and d such that cu + dv = 0. If c 鈮� 0, we can rearrange the equation to isolate u: u = (-d/c)v. This shows that u is a multiple of v. If d 鈮� 0, we can rearrange the equation to isolate v: v = (-c/d)u. This shows that v is a multiple of u. So, in either case, u or v is a multiple of the other.
04

Conclusion

We have shown that the set {u, v} is linearly dependent if and only if one of the vectors is a multiple of the other.

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