Examples from Basis and dimension

A basis of $\mathbb R^3$

Consider

The matrix with these vectors as columns row reduces to $I_3$, so its columns are linearly independent and span $\mathbb R^3$. Thus $B$ is a basis for $\mathbb R^3$.

A basis for a generated subspace

Let

The first two columns are leading columns when the matrix formed from $S$ is row reduced. They are independent and the remaining vectors are redundant, so

is a basis for $\operatorname{span}(S)$.

A basis of $P_1$

For $p=a_0+a_1x$, write

Equating coefficients gives

The coefficient matrix has determinant $-1$, so this system has a unique solution for every $a_0,a_1$. Therefore $\{1+x,x+2\}$ spans $P_1$ and is linearly independent, so it is a basis.

Standard bases and dimensions

The standard basis of $\mathbb R^2$ is

The standard polynomial basis is $\{1,x,\ldots,x^n\}$ for $P_n(\mathbb R)$. For $M_{2\times2}(\mathbb R)$, the standard matrix basis is

More generally, the standard bases give

Each count is the number of free coefficients: a polynomial in $P_n(\mathbb R)$ has $n+1$ coefficients and an $m\times n$ matrix has one for each of its $mn$ entries.

Counting prevents spanning

Two polynomials cannot span $P_2(\mathbb R)$, because

Likewise, three polynomials in $P_1(\mathbb R)$ must be linearly dependent, since