Proof of the Left Distributive Law for matrices.

[Faizan Sheikh]

Proof of the "Left Distributive Law" for matrices.

******A(B + C) = AB + AC******

Again we show that the general element of the left hand side is the same as the right hand side. We have

(A(B + C))ij = S(Aik(B + C)kj) definition of matrix multiplication

= S(Aik(Bkj + Ckj)) definition of matrix addition

= S(AikBkj + AikCkj) distributive property of the real numbers

= S AikBkj + S AikCkj commutative property of the real numbers

= (AB)ij + (AC)ij definition of matrix multiplication

where the sum is taken from 1 to k. S is the Sigma sign.
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Can somebody tell me why there is a sigma sign in the first step? Wouldn't the proof be correct without the sigma signs? How do they even derive the sigma or sums from the definition of matrix multiplication? Say we have a matrix A, and a matrix B, and say AB is defined, then the element in the first row and the first column can be computed by multiplying and adding the corresponding entries in the first row of A, and the first column of B. Now, for instance, k=1, then wouldn't AikBkj become (Ai1)(B1j), would simply mean the first column from A multplied by the corresponding entries from column 1 of B! But what is that supposed to mean? Shouldn't it have been the first row of A, and first column of B? Please help as soon as possible.

 

[krishnaprasad]

Here 'k' is not some integer. It is the number of columns in first matrix ( which is equal to number of rows in second matrix)

It means an (i,j)th element of product matrix = sum of 'k' products of (i,k)th element of first matrix and (k,j)th element of second matrix.

| 1 2 3 | | 3 0 |
| 9 0 3 | x | 2 8 |
| 7 4 |

(1,1) element = 1x3 + 2x2 + 3x7 (k=3). So he used Sigma (S) in the proof.

 

[mathwonk]

come on. this is just a big array of copies of the distributivity law for dot product.

since a(b+c) = ab + ac, where these are numbers, multiplication by oner number is linear, and since the sum of linear maps is linear, the dot product is also linear, and a matrix product is nothing but several dot products. done.


aiyee I am blinded!

 

[Hurkyl]

Can somebody tell me why there is a sigma sign in the first step?

You just did:

"then the element in the first row and the first column can be computed by multiplying and adding the corresponding entries in the first row of A, and the first column of B"