3

Show that the sum of w, x, y, z is given by:

[ w x y z ] x [ 1 ]
              [ 1 ]
              [ 1 ]
              [ 1 ]

Represent the average of w x y z in the same way

The textbook's answer

[ w x y z ] x [ 1/4 ]
              [ 1/4 ]
              [ 1/4 ]
              [ 1/4 ]

I don't even understand what the question was asking! What did it want me to do?

I think I understand that the average is represented by 1/4, but what about the part which asks for the sum. The sum = add everything together... I didnt see that occur.

Please use step-by-step reasoning =(

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1 Answer

3

Matrix multiplication between a 1x4 matrix (1 row, 4 columns) and a 4x1 matrix (4 rows, 1 column) results in a 1x1 matrix (i.e.: a scalar)

The matrix multiplication of:

[ w x y z ] x [ 1 ] = [ w(1) + x(1) + y(1) + z(1) ] = w + x + y + z
              [ 1 ]
              [ 1 ]
              [ 1 ]

Hence, the matrices multiply to equal to the sum of w, x, y, z.

If you aren't clear with matrix multiplication, a quick review can be found on Wikipedia.

To create the average, you want (w + x + y + z) divided by 4. If I write it in algebra, and expand the brackets, I get:

  • 1/4 (w + x + y + z)
  • 1/4w + 1/4x + 1/4y + 1/4z

I can easily make that happen by using that above example, but replacing all the 1's with 1/4 instead. Now I'll multiply each of w, x, y and z with 1/4 instead of 1, and I'll get the average of them!

Hope that explains it!

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Now why couldn't the textbooks be like that ^^ – asa.hoshi Jan 13 at 7:01

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