Prove that 𝐯=𝐱+i 𝐲 and ∇=𝐱-i 𝐲 are linearly independent complex vectors if and only if their rea

Prove that 𝐯=𝐱+i 𝐲 and ∇=𝐱-i 𝐲 are linearly independent...

Key Concepts

Linear Independence

Linear independence is a fundamental concept in linear algebra that dictates that a set of vectors has no nontrivial linear combination that sums to the zero vector. In other words, each vector in an independent set cannot be represented as a linear combination of the others, implying that the only solution to a linear combination equaling zero is that all coefficients are zero.

Real and Imaginary Components

Any complex vector can be decomposed into its real and imaginary parts. This separation is crucial because it allows the study of complex vectors by considering two real vectors. The linear independence of the complex vector can thus be related to the linear independence of its constituent real and imaginary parts.

Complex versus Real Vector Spaces

In complex vector spaces, scalars are complex numbers, while in real vector spaces, scalars are real numbers. Although the notion of linear independence is similar in both settings, the underlying operations in the complex field involve an additional layer of complexity due to the presence of imaginary units. Understanding this difference is key to reformulating problems from a complex context to a real one and vice versa.

Equivalence of Independence in Decomposed Forms

The equivalence between the linear independence of two complex vectors and that of their real and imaginary components hinges on the ability to separate the real and imaginary parts of any linear combination. By equating the real and imaginary components separately to zero, one can translate the condition for complex linear independence into a corresponding condition for the real vectors, thereby proving the equivalence.

Transcript

00:01 In this problem, we are told that a, b, and c are linearly independent vectors in rn.

00:08 Our first job is to show that the vectors a plus b, b plus c, and a plus c are also linearly independent.

00:17 And our second job is to determine whether that is true for the three vectors a minus b, b plus c, and a plus c.

00:27 So we will start by assuming that a, b, and c are linearly independent.

00:30 Up, and so we will review the definition of that.

00:46 This means that if we form a vector equation by taking some real numbers i, j, and k, and writing i times a plus j times b, plus k times c, and setting that equal to the zero vector, the only solution for that equation is the trivial solution where i, j, and k are all equal to zero.

01:09 So we will need to use that definition in our proof for part one.

01:15 So in part one, let's set up a similar equation, but this time for the three vectors, a plus b, b plus c, and a plus c.

01:27 So let's assume we have constants p, q, and r, so that p times a plus b plus b, plus q, times b plus c, plus r times a plus c, is the zero vector.

01:50 So we can then rewrite this in terms of the coefficients of the vector a.

01:54 So we have p plus r times vector a.

01:59 Then for vector b, its coefficients are p plus q.

02:06 And for vector c, its coefficients will be its coefficient, i should say, is q plus r.

02:14 And that's the zero vector.

02:17 Well, now notice that we've rewritten this equation in terms of the original vectors, a, b, and c.

02:24 And we said that a, b, and c are linearly independent.

02:31 That means that the only solution for this equation is the trivial solution.

02:36 That means that p plus r has to be zero, p plus q has to be zero, and q plus r has to be zero.

02:48 And now we want to use that to show that the original p, q, and r must be zero as well.

02:55 Well, let's consider the first equation and the third equation.

02:59 If we solve for r in both those equations, we have r equals minus p and r equals minus q.

03:08 So that tells us that p equals q.

03:11 But if we now consider the middle equation, if p equals q and p plus q is zero, and that means that p and q are zero.

03:21 So p is zero, q is zero, and therefore r is also zero because r is negative p or negative q.

03:32 So we have shown that p and q and r all have to be zero.

03:45 And so that means, looking back to the original equation, let's make sure you can see that, coming back up here to the original vector equation, we showed that the only solution to that for constants, pq, and r is the trivial solution, where pq and r are all zero, and that's the definition of linear independence...

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