Scalar multiplication(vectors)

1. Jan 24, 2005

C0nfused

Hi everybody,
I have a small question. I know that we have defined multiplication of a number and a vector ,for example b*A (capital letters =vectors, everything else=real numbers). We have also defined that b*(c*A)=(b*c)*A. From these two rules is a*b*c*d*...*k*Z defined (= product of n numbers with a vector) without using parentheses? What about a*b*c*E*D ? And one last thing: is scalar multiplication also written with juxtaposition? For example the above examples can be written like this: abcd...kZ and abcE*D ?
Thanks

2. Jan 25, 2005

cronxeh

You can multiply a scalar by a matrix anytime you want. However in order to multiply a matrix by another matrix their size has to be compatible.

$$A_{m x n} * B_{n x p}$$. For example,
$$A _ {2 x 3} = \left( \begin{array}{x1x2x3} 2 & 4 & 3\\ 1 & -1 & 5\\ \end{array} \right)$$ can only be multiplied by a matrix which is in $$B_{3 x n}$$ form.

So let $$B_{3x5}= \left( \begin{array}{x1x2x3x4x5} 1 & 0 & 5 & 2 & 3\\ 0 & -1 & 2 & 4 & 1\\ 4 & 5 & 6 & 7 & 8 \end{array} \right)$$

The resulting matrix will be $$A_{2x3} * B_{3x5} = C_{2x5}$$
$$C_{2x5} = \left( \begin{array}{x1x2x3x4x5} 14 & 11 & 36 & 41 & 34\\ 21 & 26 & 33 & 33 & 42\\ \end{array} \right)$$

If you have $$(a*b*c*d) * (A_{3x5} * B_{5x4} * C_{4x7}$$ The resulting matrix will be: $$(a*b*c*d) * (M_{3x7})$$

3. Jan 25, 2005

jcsd

a*b*c*E*D is not defined without first defining the multiplication of two vectors (so in otherwords you'd have to say exactly what E*D means).

It is usual to use juxtapostion for the multiplication of two scalars or the multiplication of a scalar by a vector. As there is more than one kind of product of two vectors, it's usual to use whatever binary operator denotes that product to avoid confusion.

4. Jan 26, 2005