Generalized Rank Nullity Theorem

**ThePerfectHacker** · May 18th 2008, 10:56 AM

I was wondering if there is a generalization of this theorem from finite numbers to cardinal numbers.
More specifically, let $F$ be a field and $V$ , $W$ be vector-spaces over $F$ .
Let the dimension of $V/F = \aleph_{\alpha}$ .
Let $T:V\mapsto W$ be a linear transformation then is $\mbox{dim}(\mbox{ker} T) + \mbox{dim}(\mbox{Im} T) = \aleph_{\alpha}$ ?

**NonCommAlg** · May 18th 2008, 01:18 PM

Originally Posted by ThePerfectHacker

I was wondering if there is a generalization of this theorem from finite numbers to cardinal numbers.
More specifically, let $F$ be a field and $V$ , $W$ be vector-spaces over $F$ .
Let the dimension of $V/F = \aleph_{\alpha}$ .
Let $T:V\mapsto W$ be a linear transformation then is $\mbox{dim}(\mbox{ker} T) + \mbox{dim}(\mbox{Im} T) = \aleph_{\alpha}$ ?

your question basically is whether or not we have $\dim V = \dim K + \dim(V/K),$ for any subspace $K$ of $V,$

where $\dim$ here means the cardinality of a basis of a vector space. the answer is yes! let $A=\{v_{i}: \ i \in I \}$

be a basis for $K$ and $C=\{w_j + K: \ j \in J \}$ a basis for $V/K.$ let $B=\{w_j: \ j \in J\}.$ it's easy to see that

$|B|=|C|, \ A \cap B = \emptyset,$ and $A \cup B$ is a basis for $V.$ Q.E.D.

Thread: Generalized Rank Nullity Theorem

LinkBack

Thread Tools

Display

Generalized Rank Nullity Theorem

Similar Math Help Forum Discussions

rank and nullity

Rank-nullity theorem

Using the rank-nullity theorem in a proof

The rank-nullity theorem

Rank and nullity

Search Tags