Definition: A basis of a vector space \(V\) is a set of vectors \(\{\mathbf{v}_1, \ldots, \mathbf{v}_k\}\) that is:
- Linearly independent, and
- Spans \(V\) (i.e., \(\text{span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\} = V\))
Equivalently, a basis is a minimal spanning set or a maximal linearly independent set.
Standard basis of \(\mathbb{R}^n\):
\[\mathbf{e}_1 = (1,0,\ldots,0), \quad \mathbf{e}_2 = (0,1,\ldots,0), \quad \ldots, \quad \mathbf{e}_n = (0,0,\ldots,1)\]
Example: In \(\mathbb{R}^2\), the set \(\{(1,0), (0,1)\}\) is the standard basis. But \(\{(1,1), (1,-1)\}\) is also a basis.
Key theorem: All bases of a given vector space have the same number of elements. This number is called the dimension of the space.
Coordinates: Once a basis is chosen, every vector \(\mathbf{v} \in V\) can be written uniquely as \(\mathbf{v} = c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k\). The coefficients \((c_1, \ldots, c_k)\) are the coordinates of \(\mathbf{v}\) with respect to that basis.
Children
- Definition of basis of a vector space (or subspace)A basis of $V$ is a linearly independent spanning list (or set). Every vector then has a unique finite linear...
- Definition of the standard/natural basis of R^n (or C^n)The standard basis of $\mathbb F^n$ is $(e 1,\ldots,e n)$, where $e j$ has $1$ in coordinate $j$ and zeros elsewhere....
- The standard/natural basis of R^n (or C^n) is a basis.The standard vectors span because $x=\sum jx je j$ for every $x$. They are independent because $\sum jc je j=0$...
- Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n)For an ordered basis $B=(b 1,\ldots,b n)$ of $\mathbb F^n$, set $P B=[b 1\ \cdots\ b n]$. It converts basis...
- Definition of the standard basis of the polynomials of degree at most nFor $P n(\mathbb F)$, the polynomials of degree at most $n$, the standard basis is $(1,t,t^2,\ldots,t^n)$. Every...
- Definition of the standard basis of the m by n matricesThe standard basis of $\mathbb F^{m\times n}$ consists of matrices $E {ij}$ with one entry $1$ at position $(i,j)$...
- Definition of coordinates relative to a given basisGiven an ordered basis $B=(b 1,\ldots,b n)$, the coordinate vector $[v] B=(c 1,\ldots,c n)^T$ is defined by $v=\sum...
- A set of nonzero vectors contains (as a subset) a basis for its span.A finite spanning list contains a basis for its span: repeatedly delete a redundant vector until the remaining list...
- The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.Form $A=[v 1\ \cdots\ v k]$ and find its pivot columns by row reduction. The corresponding original vectors form a...
- Each vector can be written uniquely as a linear combination of vectors from a given basis.For a basis $B=(b 1,\ldots,b n)$, every $v$ has an expansion $v=\sum ic ib i$ because $B$ spans. If another expansion...
- A set is a basis if each vector can be written uniquely as a linear combination.A finite list is a basis iff every vector in the space has exactly one linear-combination representation using that...
- CoordinatesCoordinates translate vectors into scalar lists relative to an ordered basis. For $B=(b 1,\ldots,b n)$, the...
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