Answer:
The inner product holds the commutative property while the outer product does not hold. This indicates that
and ![UV^T=(VU^T)^T](https://tex.z-dn.net/?f=UV%5ET%3D%28VU%5ET%29%5ET)
Step-by-step explanation:
For any two vectors, U and V, given as,
![U=\left[\begin{array}{c}a1\\a2\\a3\\.\\.\\an\end{array}\right] \\V=\left[\begin{array}{c}b1\\b2\\b3\\.\\.\\bn\end{array}\right]](https://tex.z-dn.net/?f=U%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Da1%5C%5Ca2%5C%5Ca3%5C%5C.%5C%5C.%5C%5Can%5Cend%7Barray%7D%5Cright%5D%20%5C%5CV%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Db1%5C%5Cb2%5C%5Cb3%5C%5C.%5C%5C.%5C%5Cbn%5Cend%7Barray%7D%5Cright%5D)
Here a1,a2,a3,b1,b2 and b3,.....,an,bn are real numbers
The products are given as
![U^TV=\left[\begin{array}{cccccc}a1&a2&a3&.&.&an\end{array}\right] \left[\begin{array}{c}b1\\b2\\b3\\.\\.\\bn\end{array}\right]\\U^TV=\left[\begin{array}{c}a1*b1+a2*b2+a3*b3+....+an*bn\end{array}\right]](https://tex.z-dn.net/?f=U%5ETV%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7Da1%26a2%26a3%26.%26.%26an%5Cend%7Barray%7D%5Cright%5D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Db1%5C%5Cb2%5C%5Cb3%5C%5C.%5C%5C.%5C%5Cbn%5Cend%7Barray%7D%5Cright%5D%5C%5CU%5ETV%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Da1%2Ab1%2Ba2%2Ab2%2Ba3%2Ab3%2B....%2Ban%2Abn%5Cend%7Barray%7D%5Cright%5D)
![V^TU=\left[\begin{array}{cccccc}b1&b2&b3&.&.&bn\end{array}\right] \left[\begin{array}{c}a1\\a2\\a3\\.\\.\\an\end{array}\right]\\V^TU=\left[\begin{array}{c}b1*a1+b2*a2+b3*a3+....+bn*an\end{array}\right]](https://tex.z-dn.net/?f=V%5ETU%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7Db1%26b2%26b3%26.%26.%26bn%5Cend%7Barray%7D%5Cright%5D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Da1%5C%5Ca2%5C%5Ca3%5C%5C.%5C%5C.%5C%5Can%5Cend%7Barray%7D%5Cright%5D%5C%5CV%5ETU%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Db1%2Aa1%2Bb2%2Aa2%2Bb3%2Aa3%2B....%2Bbn%2Aan%5Cend%7Barray%7D%5Cright%5D)
From this, it is clear that the inner product or the scalar product of two vectors U, V is commutative i.e
![U^TV=V^TU](https://tex.z-dn.net/?f=U%5ETV%3DV%5ETU)
Now for the outer or vector product
![UV^T=\left[\begin{array}{c}a1\\a2\\a3\\.\\.\\an\end{array}\right]\left[\begin{array}{cccccc}b1&b2&b3&.&.&bn\end{array}\right] \\UV^T=\left[\begin{array}{cccccc}a1*b1&a1*b2&a1*b3&.&.&a1*bn\\a2*b1&a2*b2&a2*b3&.&.&a2*bn\\a3*b1&a3*b2&a3*b3&.&.&a3*bn\\.&.&.&.&.&.\\.&.&.&.&.&.\\an*b1&an*b2&an*b3&.&.&an*bn\end{array}\right]\\UV^T=\left[\begin{array}{cccccc}a1b1&a1b2&a1b3&.&.&a1bn\\a2b1&a2b2&a2b3&.&.&a2bn\\a3b1&a3b2&a3b3&.&.&a3bn\\.&.&.&.&.&.\\.&.&.&.&.&.\\anb1&anb2&anb3&.&.&anbn\end{array}\right]](https://tex.z-dn.net/?f=UV%5ET%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Da1%5C%5Ca2%5C%5Ca3%5C%5C.%5C%5C.%5C%5Can%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7Db1%26b2%26b3%26.%26.%26bn%5Cend%7Barray%7D%5Cright%5D%20%5C%5CUV%5ET%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7Da1%2Ab1%26a1%2Ab2%26a1%2Ab3%26.%26.%26a1%2Abn%5C%5Ca2%2Ab1%26a2%2Ab2%26a2%2Ab3%26.%26.%26a2%2Abn%5C%5Ca3%2Ab1%26a3%2Ab2%26a3%2Ab3%26.%26.%26a3%2Abn%5C%5C.%26.%26.%26.%26.%26.%5C%5C.%26.%26.%26.%26.%26.%5C%5Can%2Ab1%26an%2Ab2%26an%2Ab3%26.%26.%26an%2Abn%5Cend%7Barray%7D%5Cright%5D%5C%5CUV%5ET%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7Da1b1%26a1b2%26a1b3%26.%26.%26a1bn%5C%5Ca2b1%26a2b2%26a2b3%26.%26.%26a2bn%5C%5Ca3b1%26a3b2%26a3b3%26.%26.%26a3bn%5C%5C.%26.%26.%26.%26.%26.%5C%5C.%26.%26.%26.%26.%26.%5C%5Canb1%26anb2%26anb3%26.%26.%26anbn%5Cend%7Barray%7D%5Cright%5D)
![VU^T=\left[\begin{array}{c}b1\\b2\\b3\\.\\.\\bn\end{array}\right]\left[\begin{array}{cccccc}a1&a2&a3&.&.&an\end{array}\right] \\VU^T=\left[\begin{array}{cccccc}b1*a1&b1*a2&b1*a3&.&.&b1*an\\b2*a1&b2*a2&b2*a3&.&.&b2*an\\b3*a1&b3*a2&b3*a3&.&.&b3*an\\.&.&.&.&.&.\\.&.&.&.&.&.\\bn*a1&bn*a2&bn*a3&.&.&bn*an\end{array}\right]\\VU^T=\left[\begin{array}{cccccc}b1a1&b1a2&b1a3&.&.&b1an\\b2a1&b2a2&b2a3&.&.&b2an\\b3a1&b3a2&b3a3&.&.&b3an\\.&.&.&.&.&.\\.&.&.&.&.&.\\bna1&bna2&bna3&.&.&bnan\end{array}\right]](https://tex.z-dn.net/?f=VU%5ET%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Db1%5C%5Cb2%5C%5Cb3%5C%5C.%5C%5C.%5C%5Cbn%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7Da1%26a2%26a3%26.%26.%26an%5Cend%7Barray%7D%5Cright%5D%20%5C%5CVU%5ET%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7Db1%2Aa1%26b1%2Aa2%26b1%2Aa3%26.%26.%26b1%2Aan%5C%5Cb2%2Aa1%26b2%2Aa2%26b2%2Aa3%26.%26.%26b2%2Aan%5C%5Cb3%2Aa1%26b3%2Aa2%26b3%2Aa3%26.%26.%26b3%2Aan%5C%5C.%26.%26.%26.%26.%26.%5C%5C.%26.%26.%26.%26.%26.%5C%5Cbn%2Aa1%26bn%2Aa2%26bn%2Aa3%26.%26.%26bn%2Aan%5Cend%7Barray%7D%5Cright%5D%5C%5CVU%5ET%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7Db1a1%26b1a2%26b1a3%26.%26.%26b1an%5C%5Cb2a1%26b2a2%26b2a3%26.%26.%26b2an%5C%5Cb3a1%26b3a2%26b3a3%26.%26.%26b3an%5C%5C.%26.%26.%26.%26.%26.%5C%5C.%26.%26.%26.%26.%26.%5C%5Cbna1%26bna2%26bna3%26.%26.%26bnan%5Cend%7Barray%7D%5Cright%5D)
From this, it is clear that the outer product or the vector product of two vectors U, V is not commutative i.e
![UV^T\neq VU^T](https://tex.z-dn.net/?f=UV%5ET%5Cneq%20VU%5ET)
However, it is evident that
![UV^T=(VU^T)^T](https://tex.z-dn.net/?f=UV%5ET%3D%28VU%5ET%29%5ET)