Proof on matrix multiplication

In summary: Your name]In summary, we are asked to prove that the product of a 1xm matrix C and an mxn matrix A is equal to the summation of the dot products of the rows of A with the columns of C. By rewriting the product using summation notation and comparing it to the given summation, we can see that they are equal and therefore the statement is proven.
  • #1
kolley
17
0

Homework Statement


A is an mxn matrix and C is a 1xm matrix. Prove that CA=Sum of (C sub j)*(A sub j) from j=1 to m.
Where A sub j is the jth row of A

Sorry for the messy problem statement, I couldn't figure out the summation notation on here.

Homework Equations





The Attempt at a Solution


I set (C sub j)*(A sub j)=D sub j
I then tried to manipulate it towards something, but I made no progress, thanks for any help you can give.
 
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  • #2




Thank you for your question. I am happy to assist you in proving this statement. Let's start by defining the variables:

A is an mxn matrix, which means it has m rows and n columns. We can denote the elements of this matrix as aij, where i represents the row number and j represents the column number. So A sub j would refer to the jth row of A, which can be written as (a1j, a2j, ..., amj).

C is a 1xm matrix, which means it has 1 row and m columns. We can denote the elements of this matrix as cj, where j represents the column number.

Now, let's look at the product CA. This would result in a new matrix with dimensions 1xn, since the number of columns in C must match the number of rows in A for the product to be defined. The elements of this new matrix can be denoted as (d1, d2, ..., dn), where di is the element in the ith column of the product.

We can rewrite this product using summation notation as follows:
CA = (c1, c2, ..., cm) * (a1, a2, ..., an) = (d1, d2, ..., dn)
= (c1*a1 + c2*a2 + ... + cm*an)

Now, let's compare this to the summation given in the statement:
Sum of (C sub j)*(A sub j) from j=1 to m = c1*a1 + c2*a2 + ... + cm*am

As you can see, both expressions are equal. This is because the elements of the product CA are simply the dot product of the rows of A with the columns of C. Therefore, the product CA can be written as the sum of these dot products, which is exactly what the summation is representing.

I hope this explanation helps you understand and prove the given statement. Let me know if you have any further questions.


 

1. What is the concept of proof in matrix multiplication?

The concept of proof in matrix multiplication is to show that the mathematical process used to multiply two matrices is valid and produces accurate results. This involves using logical reasoning and mathematical principles to demonstrate that the steps taken in the multiplication process are correct and the resulting product is accurate.

2. How is the proof of matrix multiplication different from other mathematical proofs?

The proof of matrix multiplication is different from other mathematical proofs because it involves the use of matrices, which are arrays of numbers, rather than just numbers or equations. This requires a different approach and understanding of matrix operations and properties.

3. Can you provide an example of a proof for matrix multiplication?

One example of a proof for matrix multiplication is the associative property, which states that the order of multiplication does not affect the result. This can be proven by setting up two matrices and showing that regardless of the order in which the multiplication is performed, the resulting product is the same.

4. Why is it important to understand the proof of matrix multiplication?

Understanding the proof of matrix multiplication is important because it allows for a deeper understanding of how matrices work and how to correctly use them in mathematical calculations. It also ensures that the results obtained from matrix multiplication are accurate and reliable.

5. Are there any common misconceptions about the proof of matrix multiplication?

One common misconception about the proof of matrix multiplication is that it is a difficult or complex concept. While it may seem intimidating at first, with practice and a solid understanding of basic matrix operations, the proof of matrix multiplication can be easily understood and applied.

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