Subset

Description

A subset of \(R_n\) is any set that contains only elements of \(R_n\). For example, \({x_0}\) is a subset of \(R_n\) if \(x_0\) is an element of \(R_n\).

Linear Combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of \(x\) and \(y\) would be any expression of the form \(ax + by\), where \(a\) and \(b\) are constants).

• By applying the linear combination over two Vectors we can represent every 2D possible Vector in the world!

Span

The linear span (also called the linear hull or just span) of a set \(S\) of vectors (from a vector space), denoted \(span(S)\), is defined as the set of all linear combinations of the vectors in \(S\).

Linear Dependent & Linear Independent

\(S = \{v_1, v_2, \dots, v_n\}\) is Linearly Dependent \(\iff\) \(c_1v_1 + c_2v_2 + \dots + c_nv_n = \mathbf{0} = \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix}\) for some \(c_i\)'s (not all) are zero (at least 1 is non-zero)

• A set of vectors is linearly dependent if there is a nontrivial linear combination of the vectors that equals 0.
• A set of vectors is linearly independent if the only linear combination of the vectors that equals 0 is the trivial linear combination (i.e., all coefficients = 0).