1. Span Verb : A Set of vectors is said to span a space if the set of all their linear combinations is the space Noun : $span(v_{1}, v_{2}) = \{ c_{1} v_{1} + c_{2} v_{2}, \forall c_{1}, c_{2} \in \mathbb{R} \}$ Note : vectors that span a space are not necessarily independent Fact : columns of $A$ span $C(A)$, special solutions of $A$ span $N(A)$ 2. Basis A Basis of a space is a set of linearly ..
1. Rank The rank of matrix in the number of pivots ($r(A) = \# $ of pivots) Also defiend as the maximum number of linearly independent columns (will be discussed later) The rank of a matrix gives in a sense the true size of the matrix. Consider an $(m \times n)$ matrix $A$ whose rank is few than $m$. While $A \mathbf{x} = \mathbf{0}$ seems to have $m$ linear equations, $m-r$ out of $m$ equations..
1. Vector Spaces A set of vectors $V$ is called a vector space if it satisfies the following axioms $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}, \mathbf{v} \in V$ $(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w}), \forall \mathbf{u} \mathbf{v} \mathbf{w}$ There exists a zero vector $\mathbf{0} \in V$, such that $\mathbf{v} + \mathbf{0} = \mathbf{v}, \forall \mathbf{v}..