91Ó°ÊÓ

A vector space is a fundamental concept in linear algebra that deals with a set of vectors. These vectors can be added together and scaled by numbers, called scalars, typically from the real numbers (\( \mathbb{R} \)). To verify if a set forms a vector space, it must satisfy certain criteria:

• Closure under addition and scalar multiplication: Adding any two vectors in the set or multiplying a vector by a scalar should produce another vector in the set.
• Existence of the zero vector: The set should contain a vector where all components are zero.
• Associativity and commutativity of addition: It must be indifferent which order vectors are added.

By ensuring these conditions are met, one can verify if a given set forms a vector space.

Linear independence is a key concept in understanding vector spaces and subspaces. A set of vectors is said to be linearly independent if no vector in the set can be written as a combination of the others.

• Mathematically, this means that if you have a set of vectors \( \mathbf{v}_1, \mathbf{v}_2, \, \ldots, \mathbf{v}_n\), the only solution to \( c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \, \ldots, + c_n \mathbf{v}_n = 0\) is \( c_1 = c_2 = \, \ldots, = c_n = 0\).

For example, in the exercise solution, the vectors:
\( \mathbf{b}_1 = \left[ \begin{array}{c} 2 \ -3 \ 2 \ 1 \end{array} \right], \mathbf{b}_2 = \left[ \begin{array}{c} 6 \ -9 \ 6 \ 3 \end{array} \right], \mathbf{b}_3 = \left[ \begin{array}{c} 7 \ -12 \ 6 \ 3 \end{array} \right] \)
are tested for linearly independence by solving \( c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 + c_3 \mathbf{b}_3 = \mathbf{0} \). If all coefficients \(c_i\) are zero for this solution, they are independent.
This concept is foundational for determining bases and dimensions of subspaces.

The dimension of a subspace is an important property describing the subspace's structure. It is the number of vectors in the basis of the subspace, which can fully span the subspace without redundancy.
For instance, if we have 3 linearly independent vectors:
\( \mathbf{b}_1 = \left[ \begin{array}{c} 2 \ -3 \ 2 \ 1 \end{array} \right], \mathbf{b}_2 = \left[ \begin{array}{c} 6 \ -9 \ 6 \ 3 \end{array} \right], \mathbf{b}_3 = \left[ \begin{array}{c} 7 \ -12 \ 6 \ 3 \end{array} \right] \)
and they span a subspace, then the dimension of that subspace is 3.

• The dimension essentially tells us how many directions are necessary to describe any vector in the subspace fully.
• This number helps in understanding the complexity and structure of the subspace within the larger vector space.

A basis of a subspace is a set of vectors that are both linearly independent and span the subspace. In other words, any vector in the subspace can be expressed as a linear combination of the basis vectors.
For the subspace given by:
\( S = \left\{ \left[ \begin{array}{c} 2u + 6v + 7w \ -3u - 9v - 12w \ 2u + 6v + 6w \ u + 3v + 3w \end{array} \right] : u, v, w \in \mathbb{R} \right\} \),
the basis vectors found were:
\( \mathbf{b}_1 = \left[ \begin{array}{c} 2 \ -3 \ 2 \ 1 \end{array} \right], \mathbf{b}_2 = \left[ \begin{array}{c} 6 \ -9 \ 6 \ 3 \end{array} \right], \mathbf{b}_3 = \left[ \begin{array}{c} 7 \ -12 \ 6 \ 3 \end{array} \right] \)
which span \( S \).

• To determine a basis, identify the minimum number of vectors needed to span the subspace while ensuring they are linearly independent.
• This process typically involves expressing subspace vectors in terms of parameters and then confirming their independence through methods like row-reducing to echelon form in a matrix.