Vectors play a big role in understanding linear systems. They help us visualize and make sense of solutions to these systems. In university-level algebra, we learn to express linear systems using vectors. This makes calculations easier and helps us see the geometric connections between solutions.

So, what exactly is a linear system? A linear system usually looks like a set of equations like this:

$$\begin{align*} a_1x_1 + a_2x_2 + \ldots + a_nx_n &= b_1 \\ c_1x_1 + c_2x_2 + \ldots + c_nx_n &= b_2 \\ &\vdots \\ k_1x_1 + k_2x_2 + \ldots + k_nx_n &= b_m \end{align*}$$

Here, $x_1, x_2, ..., x_n$ are the values we want to find, and $a_i, b_i$ are numbers that stay constant.

From a vector viewpoint, we can simplify this system. We can create a matrix (which is just a grid of numbers) called $A$ for the coefficients, a vector called $\mathbf{x}$ for the variables, and another vector called $\mathbf{b}$ for the constants. This allows us to write the system as:

$$A\mathbf{x} = \mathbf{b}$$

This way of writing it makes solving linear systems easier. There are different methods like Gaussian elimination and matrix inversion that we can use. But more than just being easier to work with, vectors give us a way to think about the solutions geometrically.

Imagine each equation in our system as a "flat" surface (called a hyperplane) in a higher-dimensional space. The solution to the linear system will be where these hyperplanes intersect or cross each other. This gives us a better understanding of what the solutions look like.

There are three scenarios we can have:

1. Unique Solutions: If the hyperplanes meet at one specific point, we have one solution. This happens when the equations are independent and make sense together.

2. No Solutions: If the hyperplanes are parallel and don't meet at all, there are no solutions. This is when the equations contradict each other.

3. Infinite Solutions: If the hyperplanes meet along a line or a flat surface, there are endless solutions. This usually happens when the equations are dependent, meaning some don't bring in new information.

Vectors are crucial in these situations. Each point in our space can be shown with a vector, making it easier to visualize the solutions. If we find intersection points, we can see them as combinations of vectors that lead back to the vectors forming the solution set.

Now, let's talk about how vectors help us understand concepts like linear dependence and independence. A group of vectors, like $\{\mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_k}\}$, is called linearly independent if the only way to combine them to equal zero (the zero vector) is if all their coefficients (the $c$'s) are zero:

$$c_1\mathbf{v_1} + c_2\mathbf{v_2} + \ldots + c_k\mathbf{v_k} = \mathbf{0}$$

This means all the vectors add something unique to the set. If we have dependent vectors, at least one can be written to combine from the others, which reduces the number of dimensions we’re working with. It tells us that some equations aren’t giving us new info, so we have fewer dimensions to look for solutions.

There are many real-world uses for vectors. Take optimization problems, for example. In linear programming, we can use vectors to describe problems and find helpful solutions graphically or with matrices.

Additionally, computer graphics use vectors a lot. When we rotate, move, or resize images, we're using vectors to represent these changes. This math helps create a 3D world in films and video games.

Vectors are also important in data science and machine learning. For instance, in Principal Component Analysis (PCA), we use vectors to help simplify data and discover patterns. The geometry of high-dimensional spaces shown by vectors helps us solve complex problems and understand different data.

In summary, vectors help us see and understand solutions to linear systems more clearly. They turn complicated equations into more visual forms. This understanding benefits many fields, including physics, engineering, computer science, and economics.

As we keep looking into linear systems with vectors, we discover a web of connections and uses. It shows that these mathematical ideas are not just abstract but real tools for understanding the world around us. The relationship between vectors and linear systems is a fundamental part of math that impacts many areas.