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Linear independence is a fundamental concept in vector spaces, crucial for understanding vector decomposition. A set of vectors is said to be linearly independent if no vector in the set can be written as a linear combination of the others. In simpler terms, each vector provides a unique direction in the vector space, and none of them is redundant.

To visualize this, imagine having two vectors in a plane. If these vectors are linearly independent, they point in different directions and are not a mere scaling of one another. Now, in the exercise, the vectors \(\begin{bmatrix} 1 \ 1 \ 0 \end{bmatrix}\), \(\begin{bmatrix} 1 \ 0 \ 1 \end{bmatrix}\), and \(\begin{bmatrix} 0 \ 1 \ 1 \end{bmatrix}\) are linearly independent. None can be expressed as a combination of the other two, meaning they span a 3-dimensional space, which is necessary for any vector in \(\mathbb{R}^3\) to be expressed as a linear combination of these vectors.

Knowing that a set of vectors is linearly independent helps us confirm that we can find coefficients \(\alpha, \beta, \gamma\) to express any vector \(\begin{bmatrix} p \ q \ r \end{bmatrix}\) using these vectors.

A vector space is a collection of objects called vectors, which can be added together and multiplied by scalars. These spaces, also known as linear spaces, follow specific rules or axioms, such as closure under addition and scalar multiplication, existence of a zero vector, etc. This means that if you add two vectors or scale a vector within the vector space, you will still be within the vector space.

The vectors \(\begin{bmatrix} 1 \ 1 \ 0 \end{bmatrix}\), \(\begin{bmatrix} 1 \ 0 \ 1 \end{bmatrix}\), and \(\begin{bmatrix} 0 \ 1 \ 1 \end{bmatrix}\) form a vector space in this exercise. When combined linearly — by scaling and adding — these vectors can produce any vector in \(\mathbb{R}^3\). This ability to generate or 'span' the entire space of \(\mathbb{R}^3\) from these three vectors is possible because they are linearly independent, forming a basis for the vector space.

Understanding vector spaces helps us grasp how vectors can interact within a geometric framework and gives us the tools to manipulate vectors in multidimensional spaces.

Linear combinations involve creating a new vector by scaling and adding a set of vectors. This comes into play when expressing one vector as a combination of others, which is a core ability in vector decomposition.

In the original exercise, we were tasked with expressing \(\begin{bmatrix} p \ q \ r \end{bmatrix}\) as a linear combination of the vectors \(\begin{bmatrix} 1 \ 1 \ 0 \end{bmatrix}\), \(\begin{bmatrix} 1 \ 0 \ 1 \end{bmatrix}\), and \(\begin{bmatrix} 0 \ 1 \ 1 \end{bmatrix}\). The expression \(\alpha \begin{bmatrix} 1 \ 1 \ 0 \end{bmatrix} + \beta \begin{bmatrix} 1 \ 0 \ 1 \end{bmatrix} + \gamma \begin{bmatrix} 0 \ 1 \ 1 \end{bmatrix}\) represents this linear combination. Each constant \(\alpha\), \(\beta\), and \(\gamma\) represents how much of each vector contributes to the final result.

Seeing a vector as a linear combination of others not only aids in vector decomposition but also is crucial in data analysis, physics, computer graphics, and many other fields. By mastering linear combinations, you're equipped to solve complex problems involving vector transformation and manipulation.