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Vectors are considered to be linearly independent if there is no way to write one of them as a linear combination of the others. In simpler terms, no vector in the group can be created by scaling and adding the other vectors. For example, in a two-dimensional plane, two vectors are linearly independent if they are not on the same line. Picture having two arrows: if one arrow cannot be stretched or shrunk to overlay the second arrow, they're independent.

It's like having different ingredients in a recipe — if each one brings a unique flavor, they are independent. But if one is just a pinch of salt more than the other, they're not offering anything new. In the case of vectors, if we can't use multiplication or addition to replicate any vector in the set from the others, they are linearly independent.

Redundant vectors, in contrast to linearly independent ones, do not provide any new information to a set of vectors. They are the 'extra' ingredients that don't change the overall taste of the recipe. They are like echoes in the world of vectors — merely repeating the direction or magnitude already provided by another vector in the set.

When identifying redundancy, visualize whether removing a vector from a set changes your ability to span the space. If removal makes no difference, the vector was just a repeat, and hence, redundant. In the example of \(\left[\begin{array}{r}7 \ 11\end{array}\right], \left[\begin{array}{l}0 \ 0\end{array}\right]\), there is no loss of information if the zero vector is omitted, signifying its redundancy in the set.

A scalar multiple is essentially the result of multiplying a vector by a number, which is called a scalar. This operation changes the magnitude of the vector but not its direction. Think of a scalar multiple as adjusting the volume on your music player — the song doesn't change, just how loud it is. Mathematically, if you take a vector \(\mathbf{v}\) and a scalar \(k\), the scalar multiple is \(k\mathbf{v}\).

For instance, if we have a vector \(\left[\begin{array}{r}2 \ 3\end{array}\right]\) and we apply a scalar multiple of 4, the new vector would be \(\left[\begin{array}{r}8 \ 12\end{array}\right]\), which is still the same 'arrow' or direction, just longer in length.

The zero vector is the equivalent of zero in the land of vectors. Represented by \(\left[\begin{array}{l}0 \ 0\end{array}\right]\) in two dimensions, it's like a blank slate — it holds no direction and no magnitude. A zero vector is like being at a crossroads with no signposts; there's nowhere to go because it points to nothing.

If you include a zero vector in a set, it automatically makes the set linearly dependent. In terms of linear span, the zero vector contributes nothing — it's like adding water to a soup; it doesn't change the essential flavors. However, its presence is critical in concepts like vector spaces where it serves as the identity element for vector addition.