23.10. Exercises

1. Determine whether the following vectors are linearly independent: [ 2 ] [1] [3] ⃗v1 = 1 , ⃗v2 = 0 , ⃗v3 = 1 [1] [2] [3] −1 1 0

This exercise involves checking if three vectors are linearly independent. To do this, we can arrange them as rows in a matrix and perform calculations. The goal is to see if the only way to express the zero vector is by setting all coefficients in a linear combination to zero. If we can create a non-trivial solution (where coefficients are not all zero), then the vectors are dependent.

Imagine three ropes (the vectors) attached to a ring. If you can pull on them in such a way that they all cancel out to just one point (the zero vector), the ropes are independent. But if you can express one rope as a combination of two others (like if one rope is just a part of another), then they are dependent, just like in our exercise.

2. Show that the set {(1,2),(2,4)} is linearly dependent.

Here, we are tasked with showing that these two vectors are linearly dependent. This means we can express one vector as a multiple of the other. Notice that (2,4) is simply 2 times (1,2). Because one vector can be written as a combination of the other, they are dependent.

Think of two paths on a map—if one is just a straight line that runs along the other, they represent the same direction. Here, the second path doesn’t add new information; it’s just a stretched version of the first. Similarly, one vector here represents the same direction as the other, making them dependent.

3. For what values of a is the set {(1,2),(a,1)} linearly independent?

To find the values of a that keep these vectors independent, we look for a scenario where the second vector can't be expressed as a multiple of the first. The condition everyone must avoid is when a vector becomes a scaled version of another. Setting up a relationship such as 1 = k(2) where k is some constant can help us find the boundaries of the values we can use for 'a'. Solving this will provide values where the vectors are always independent.

Imagine two friends who cannot be considered the same person in a photo. If one friend stands next to the other in a similar pose at a certain distance, they might look like a double. If we change how close or far apart they stand (this represents 'a'), we can ensure they are clearly two distinct individuals, just as we want our vectors to remain unique.

4. In R4, is it possible for 5 vectors to be linearly independent? Justify your answer.

In R4, we are working in four-dimensional space. Any set of vectors exceeding the space's dimension will inevitably become dependent because you can't fit more than four distinct directions in a four-dimensional framework. Thus, five vectors in R4 cannot be independent; there will be redundancy or overlap among them.

Picture a four-dimensional box that can hold only four distinct Lego stacks. If you attempt to add a fifth stack, it won't fit without overlapping or stacking directly over one of the existing four. Just like that, you can't add more linearly independent vectors than the dimensions allow.

5. Using Gaussian elimination, check the independence of: ⃗v1 = (1,2,3,4), ⃗v2 = (0,1,1,0), ⃗v3 = (1,3,4,4)

To check if these vectors are independent, we form a matrix from them and use Gaussian elimination to achieve row echelon form. We will count the number of pivot columns. If the number of pivot columns equals the number of vectors, the set is independent. If not, then it is dependent.

Consider three paths leading to the same point—a conference room. If all paths remain separate and distinct without overlapping (pivot columns), then each represents a unique way to the room (independent). However, if any paths overlap or simply lead to the same direction, it means some paths are redundant (dependent).

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