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Mathematics-Online course: Linear Algebra - Matrices - Linear Maps

The dimensions of Image and Kernel

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Let $ \alpha: V\longmapsto W$ be a linear map and let $ \operatorname{dim} V<\infty$. Then the following holds true:
(i)
$ \operatorname{Ker}(\alpha)$ is a subspace of $ V$.
(ii)
$ \operatorname{Im}(\alpha)$ is a subspace of $ W$.
(iii)
$ \operatorname{dim} V = \operatorname{dim}\operatorname{Ker}(\alpha) + \operatorname{dim}\operatorname{Im}(\alpha)$

The first two assertions follow directly from the linearity of $ \alpha$.

To prove the identity for the dimensions choose a basis $ E'=\{e_1,\ldots,e_m\}$ of $ \operatorname{Ker}\alpha$ and complete it by $ \{e_{m+1},\ldots,e_n\}$ to a basis $ E$ of $ V$. Since

$\displaystyle \alpha(\sum_{i=1}^n v_i e_i) = \sum_{i=m+1}^n v_i \alpha(e_i) , $

we have

$\displaystyle \operatorname{Im}\alpha = \operatorname{span} F, \quad F=\{\alpha(e_{m+1}),\ldots, \alpha(e_n)\}\, . $

Further, $ F$ is linearly independent. It is even a basis of $ \operatorname{Im}\alpha$, since from

$\displaystyle \sum_{i=m+1}^n v_i \alpha(e_i) = 0 $

it follows that $ \sum_{i=m+1}^n v_i e_i\in\operatorname{Ker}\alpha$ and,consequently, $ v_i=0$. Assertion (3) now follows from comparing the numbers basis vectors.
(Authors: Burkhardt/Höllig/Hörner)
To illustrate the dimension formula, let us consider the $ k$-th derivative on the space of polynomials of degree $ \le n$.

At first, let $ k=1$ and $ n=2$. For the space of polynomials of degree $ \le 2$ the monomials $ \{1,x,x^2\}$ form a basis. Hence, the space has the dimension 3.

For $ p(x)=a_0+a_1x+a_2x^2$ the polynomial $ p'(x)=a_1+2a_2x$ has degree $ \le 1$. Thus, the image space has dimension 2.

The derivative of a constant vanishes, and the constants form a one-dimensional subspace. Hence, the kernel of the mapping has dimension 1 and the dimension formula is satisfied by $ 3=1+2$.

In general a polynomial has the form

$\displaystyle p(x)=\sum_{l=0}^n a_lx^l $

and its $ k$-th derivative has the form

$\displaystyle p^{(k)}(x)=\sum_{l=k}^n \frac{l!}{(l-k)!}a_lx^{l-k}\,. $

Hence, the image is a polynomial of degree $ \le n-k$ and polynomials of degree $ <k$ are nullified. Thus, the dimension formula reads as follows:

$\displaystyle n+1=\underbrace{((n-k)+1)}_{\dim \operatorname{Im}} + \underbrace{k}_{\dim \operatorname{Ker}}\,. $

(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005