Scalar multiplication(vectors)

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In summary, the conversation discusses the rules for multiplication of a number and a vector, as well as the use of parentheses and juxtaposition in scalar multiplication and matrix multiplication. The conversation also mentions the importance of defining the multiplication of two vectors in order to determine the product of a sequence of numbers and a vector.
  • #1
C0nfused
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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
 
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  • #2
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.

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

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

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

If you have [tex](a*b*c*d) * (A_{3x5} * B_{5x4} * C_{4x7}[/tex] The resulting matrix will be: [tex](a*b*c*d) * (M_{3x7})[/tex]
 
  • #3
C0nfused said:
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

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
Thanks for your answers
 

1. What is scalar multiplication in vectors?

Scalar multiplication in vectors is the process of multiplying a vector by a scalar, which is a single number. This results in a new vector that is parallel to the original vector but has a different magnitude.

2. What is the difference between scalar and vector multiplication?

The main difference between scalar and vector multiplication is that scalar multiplication involves multiplying a vector by a single number, while vector multiplication involves multiplying two vectors together to create a new vector.

3. How is scalar multiplication represented mathematically?

Scalar multiplication is represented as c * v, where c is the scalar and v is the vector. This means that each component of the vector is multiplied by the scalar, resulting in a new vector with the same direction but a different magnitude.

4. What is the purpose of scalar multiplication in vectors?

Scalar multiplication is used to scale or resize a vector. It can also be used to find a vector that is parallel to the original vector but has a different magnitude. This is useful in many mathematical and scientific applications.

5. Can scalar multiplication change the direction of a vector?

No, scalar multiplication does not change the direction of a vector. It only changes the magnitude of the vector. In order to change the direction of a vector, vector addition or subtraction must be used.

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