91Ó°ÊÓ

A basis of a vector space is a set of vectors that both spans the vector space and is linearly independent. This means that any vector in the vector space can be expressed as a linear combination of the basis vectors. Furthermore, none of the basis vectors is redundant as no vector can be formed from the others.
For example, in the context of \(\mathbb{R}^{4}\), a set of four vectors is needed to form a basis. This is because \(\mathbb{R}^{4}\) is four-dimensional, requiring four vectors that can cover each dimension fully while being independent of each other.
Extending a smaller set, like \(T = \{(1,2,3,0),(2,3,4,1),(3,4,5,3)\}\), to a basis involves finding additional vectors that maintain these properties. It ensures any vector in \(\mathbb{R}^{4}\) can be made from this set.

Gaussian elimination is a method used to solve systems of linear equations, find the rank of a matrix, and determine linear independence of vectors. It involves row operations to transform a matrix into its row-echelon form or reduced row-echelon form.
In our exercise, the process begins by taking the matrix formed by the vectors in the set \(T\) as rows:

\[\begin{bmatrix} 1 & 2 & 3 & 0 \ 2 & 3 & 4 & 1 \ 3 & 4 & 5 & 3 \end{bmatrix}\]

The goal is to perform row operations until it is clear if the matrix rows are independent. Row reduction techniques like swapping rows, scaling rows, and replacing rows, help in simplifying the matrix. If after reduction, there are no zero rows or redundant equations, the vectors are linearly independent.

The standard basis vectors are a set of vectors that are particularly simple and form a basis for vector spaces like \(\mathbb{R}^{n}\). Each vector in this set has a 1 in one position and 0s elsewhere.
For example, in \(\mathbb{R}^{4}\), these vectors are:

\(e_1 = (1, 0, 0, 0)\)
\(e_2 = (0, 1, 0, 0)\)
\(e_3 = (0, 0, 1, 0)\)
\(e_4 = (0, 0, 0, 1)\)

These vectors are independent and together span the entire space. In the exercise, the vector \(e_4 = (0, 0, 0, 1)\) was chosen to extend the set \(T\) to a basis. Because it's not in the span of the other vectors in \(T\), it maintains linear independence.

A linear combination involves creating new vectors by multiplying each vector in a set by a scalar and summing the results. It is a foundational concept in linear algebra, helping to express vectors as combinations of others.
For example, given vectors \(v_1\), \(v_2\), and \(v_3\), a linear combination would be expressed as:

\(a \, v_1 + b \, v_2 + c \, v_3\)

where \(a\), \(b\), and \(c\) are scalars.
In our exercise, no vector in \(T\) can be written in terms of the others as a linear combination, confirming their linear independence. Extending the set with another independent vector creates a full basis, capturing the essence of expressing any vector in \(\mathbb{R}^{4}\) as a complete linear combination of the basis vectors.