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A basis in linear algebra is a set of vectors that can represent every vector in a given vector space through a linear combination. In simpler terms, if you think of a vector space as a stage, the basis vectors are the essential actors needed to perform every possible scene.

For a set of vectors to be a basis, they must satisfy two conditions:

• They must be linearly independent, meaning no vector in the set can be derived from a combination of others in the same set.
• They must span the vector space, so any vector in the space can be expressed as a combination of these basis vectors.

In our exercise, vectors \(v_1 = (1,0,2)\) and \(v_2 = (0,1,1)\) are part of the basis for \(\mathbb{R}^3\). To complete the basis, another vector \(v_3 = (1,1,1)\) is added, resulting in the new basis \(\{(1,0,2),(0,1,1),(1,1,1)\}\) for \(\mathbb{R}^3\). This set can span the entire space and is minimal as per the space's dimension.

Linear independence is a key concept in linear algebra. It's about ensuring that no vector in a set is redundant. Vectors are linearly independent if no vector can be written as a linear combination of others in the set.

For example, in our problem, to prove that vectors \(v_1\) and \(v_2\) are linearly independent, we check that there's no scalar multiple that can make one vector look like the other. If \(c_1v_1 + c_2v_2 = 0\) leads only to the trivial solution where \(c_1 = 0\) and \(c_2 = 0\), then the vectors are independent.
This condition is crucial for forming a basis because it prevents any vector in the basis from being expressed as a combination of the others, ensuring that each vector adds unique information.

A vector space is a fundamental structure in linear algebra. It's a collection of vectors where you can add any two vectors and multiply a vector by a scalar, and the results will still be within the same space.

Vector spaces must satisfy specific properties:

• Closure under addition: The sum of any two vectors in the space remains within the space.
• Closure under scalar multiplication: Any scalar multiple of a vector in the space remains within the space.
• It contains a zero vector.

In \(\mathbb{R}^3\), the vector space consists of all 3-dimensional vectors. A basis for this space, such as \(\{(1,0,2), (0,1,1), (1,1,1)\}\), allows you to express any vector in \(\mathbb{R}^3\) using these three as a foundation. This makes understanding vector spaces essential for delving deeper into the concepts of dimension and span.