Adding 2 vectors definition Springer Text Page 11 of Matrix Algebra by James E Gentle states 
"Two vectors can be added if they have the same number of elements."
A 3 x 4 matrix and 4 x 3 will have same no. of elements (12) so can they be added even when corresponding elements are missing? How is it?
Thanks
Siju
 A: First of all, I'm not sure what definition of vector that book uses. From the sentence "Two vectors can be added if they have the same number of elements." I'm going to just say yikes that's probably a terrible definition of a vector.
Despite that, the answer to your question is no. Not really. Operations in mathematics are "typed" in a sense. That is if vectors have "type" $V$, $+$ has type $V\times V\to V$, (where the type $A\to B$ is the type of functions from $A$ to $B$, and the type $V\times V$ is the type of pairs of things of type $V$). I.e., addition takes two elements of the same "type" and produces a third thing of the same "type." 
Now $3\times 4$ matrices, $4\times 3$ matrices, and vectors of length 12 all are different types, even though they have the same, well the proper word is dimension, not number of elements. Thus you can't add $3\times 4$ matrices to $4\times 3$ matrices or to vectors of length 12, because they aren't the same type, and addition requires the things it adds to be of the same type.
Side note: The reason I have the word "type" in quotation marks above is because under standard foundations of mathematics, the correct word here is "set" rather than "type." I would ordinarily use the proper mathematical language, but it seems that this book doesn't, and I'd rather be understood than be completely precise. 
