Linear algebra problem with transpose

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SUMMARY

The formula for the transpose of the product of matrices and a vector, specifically (ABx)T, can be derived using the property (AB)T = BTAT. This leads to the conclusion that (ABx)T = xTBTAT, which simplifies the expression without extensive summation notation. The discussion emphasizes the importance of understanding matrix multiplication properties to derive formulas involving multiple matrices and vectors. The participants suggest that a cleaner approach exists, avoiding complex summation methods previously encountered.

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  • Understanding of matrix multiplication and properties
  • Familiarity with matrix transpose operations
  • Basic knowledge of linear algebra concepts
  • Ability to manipulate vectors and matrices
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  • Learn about the implications of matrix multiplication on vector transformations
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Homework Statement



Find a formula for (ABx)T, where x is a vector and A and B are matrices of appropriate sizes.

Homework Equations



(AB)T = BTAT among a few others, probably the most relevant one with transposes here.

The Attempt at a Solution



I'm wondering what this "formula" the problem asks for looks like. I know that (ABx)T = xTBTAT but I can't really see how much simpler another formula might be.

Any ideas what route to take to find a formula for this, preferable without the use of a lot of summation notation or anything beyond a beginning linear algebra course? We used a lot of summations earlier trying to tackle this problem and it just looked messy; I'm hoping there may be a cleaner way of approaching the problem.
 
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Perhaps what the question is getting at is for you to use the formula (AB)T = BTAT which you know works for a product of two conformable matrices and get the formula for three of them from that, being careful about properties of matrix multiplication that you know.
 
I got it for three matrices and that was before using all the sums. We didn't really go back to the problem again (she said it was so ugly anyway) so I guess that's it for the problem. :) Thanks
 

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