Problem 49 - Differential Equations and Linear Algebra

Question

Prove that if \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\}\) is linearly independent and \(\mathbf{v}_{3}\) is not in \(\operatorname{span}\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\},\) then \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\\}\) is linearly independent.

Step-by-Step Solution

Answer

To prove that \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) is linearly independent given that \(\{ \mathbf{v}_1, \mathbf{v}_2\}\) is linearly independent and \(\mathbf{v}_3\) is not in their span, we first consider a linear combination of the vectors equal to the zero vector: \(c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + c_3\mathbf{v}_3 = \mathbf{0}\). If \(c_3 = 0\), then it follows that \(c_1 = c_2 = 0\) due to the linear independence of \(\{ \mathbf{v}_1, \mathbf{v}_2\}\). If \(c_3 ≠ 0\), we rearrange the equation to find \(\mathbf{v}_3\) in the span of \(\{ \mathbf{v}_1, \mathbf{v}_2\}\), which contradicts our assumption that \(\mathbf{v}_3\) is not in their span. Thus, \(c_3\) must also be zero. Therefore, \(c_1 = c_2 = c_3 = 0\), and the set \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) is linearly independent.

Step 1: Define a linear combination for the vectors

Let \(c_1\), \(c_2\), and \(c_3\) be scalar coefficients such that: \[ c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + c_3\mathbf{v}_3 = \mathbf{0} \]

Step 2: Rearrange the equation

Rearrange the equation for \(\mathbf{v}_3\). We get: \[ \mathbf{v}_3 = -\frac{c_1}{c_3}\mathbf{v}_1 - \frac{c_2}{c_3}\mathbf{v}_2 \] Now, we need to prove that \(c_1 = c_2 = c_3 = 0\).

Step 3: Consider the cases when \(c_3 = 0\) and \(c_3 ≠ 0\)

If \(c_3 = 0\), then the above equation becomes: \[ \mathbf{0} = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 \]

Step 4: Prove that \(c_1 = c_2 = 0\) when \(c_3 = 0\)

Since \(\{ \mathbf{v}_1, \mathbf{v}_2\}\) is linearly independent, then the only way for \(\mathbf{0} = c_1\mathbf{v}_1 + c_2\mathbf{v}_2\) is if \(c_1 = c_2 = 0\).

Step 5: Prove that \(c_1 = c_2 = c_3 = 0\) when \(c_3 ≠ 0\)

If \(c_3 ≠ 0\), then the LHS of the rearranged equation in Step 2 is not zero, and the above equation tells us that \(\mathbf{v}_3\) is in the span of \(\{ \mathbf{v}_1, \mathbf{v}_2\}\). However, given that \(\mathbf{v}_3\) is not in their span, we must have a contradiction. Thus, it must be the case that \(c_3 = 0\).

Step 6: Conclusion

Since \(c_1 = c_2 = c_3 = 0\), the set \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) is linearly independent.

Key Concepts

Vector Spaces

A vector space is a collection of vectors that can be scaled and added together following specific rules. These vectors can be thought of as arrows pointing in different directions. They can be added together and multiplied by numbers (called scalars) to create new vectors within the same space.

Think of a simple example of a vector space as a flat plane where each point is represented by a vector. This plane is filled with infinitely many vectors in all possible directions, but they all follow the rules of addition and scalar multiplication.

• Vector addition means combining two vectors to yield a third vector in the same space.
• Scalar multiplication involves multiplying a vector by a number, shrinking or stretching the vector.

Understanding the concept of vector spaces is crucial in studying linear algebra, as it forms the foundation for exploring more complex structures such as linear transformations and inner product spaces.

Linear Combinations

Linear combinations are a way of constructing new vectors from a set of given vectors. Imagine you have a bag of building blocks—each block is a vector. By adding and scaling these blocks, you can create new structures, or in mathematical terms, new vectors. The expression of a linear combination looks like this:

\[ a_1\mathbf{v}_1 + a_2\mathbf{v}_2 + \ldots + a_n\mathbf{v}_n \]
where \(a_1, a_2, \ldots, a_n\) are scalars, and \(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\) are vectors.

This concept is essential because it helps us determine whether a vector can be constructed from a given set, thereby linking to the idea of the span of vectors.

• If a vector can be represented as a linear combination of others in a set, it lies within their span.
• Solving systems of linear equations often involves finding suitable linear combinations.

By mastering linear combinations, you'll be better equipped to understand vector spaces' structure and relationships.

Span of Vectors

The span of a set of vectors is like the family of all possible vectors you can create through linear combinations of those vectors. Imagine having a set of vectors as the ingredients for a recipe. The span is all the possible dishes (vectors) you can make from those ingredients.

Formally, the span of vectors \([\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n]\) is the collection of all vectors that can be written as:
\[ c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \ldots + c_n\mathbf{v}_n \]
where \(c_1, c_2, \ldots, c_n\) are any scalars.

The span is crucial because it tells you about the reach of a vector set—what other vectors can be formed from them. Understanding the span allows you to determine linear independence, a key concept in the given exercise.

• If \(\mathbf{v}_3\) cannot be expressed as a linear combination of \(\mathbf{v}_1\) and \(\mathbf{v}_2\), it lies outside their span.
• The notion of span leads directly into the idea of a basis, which provides a minimal set of vectors to describe a vector space.

Exploring the span gives vital insights into the dimensions and possibilities within a vector space.