Basis

Topics

• Dual Space
• Dual Vector

Notes

• The basis of a span is the smallest linearly independent list of vectors that constitute the span
• $\text{span}({\vec{v}}_1,\dots ,{\vec{v}}_m)=\{c_1{\vec{v}}_1+\dots +c_m{\vec{v}}_m:c_1,\dots ,c_m\text{ are scalars}\}$
  • The span of some vectors is the set of all possible linear combinations of the vectors
• $V\in {\mathbb{R}}^n\to \text{dim}(V)=m\le n$
  • $\text{dim}(V)$ is the number of vectors in the basis of $V$
    1. There are at most $m$ linearly independent vectors in $V$
    2. We need at least $m$ vectors to span $V$
    3. If $m$ vectors in $V$ are linearly independent, they form a basis of $V$
    4. If $m$ vectors in $V\text{ span }V$, then they form a basis of $V$
• $\text{dim(im(}A))=\text{rank}(A)$
  • The basis of the image of a matrix is the set of the columns containing the leading variables. Can be found by taking the $\text{rref}(A)$ and then looking at the corresponding columns with a leading 1 (can’t look at the matrix from $\text{rref}(A)$ specifically, have to look at $A$)
  • $\text{dim(ker(}A))=m-\text{rank}(A)=\text{nullity}(A)$
• The basis of the kernel of matrix is the set of the columns containing the free variables. Can be found by taking the $\text{rref}(A)$, finding $\vec{x}$ in terms of the free variables and then taking the vectors that arise as a result of isolating the free variables as the basis vectors
  • $\text{nullity}(A)+\text{rank}(A)=m$
  • Called the Rank-nullity theorem
  • Is a portion of the fundamental theorem of linear algebra