Basis

#Math

Topics

The basis of a span is the smallest linearly independent list of vectors that constitute the span
$\text{span}(\vec v_1,\ldots,\vec v_m)={c_1\vec v_1+\ldots+c_m\vec v_m:c_1,\ldots,c_m\text{ are scalars}}$
- The span of some vectors is the set of all possible linear combinations of the vectors
$V\in\mathbb{R}^n\rightarrow\text{dim}(V)=m\le n$
- $\text{dim}(V)$ 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