Can A Tensor Multiplied by 0 Produce a Null Tensor?

[tenchotomic]

Mutiplication of a scalar by a 0 gives 0(scalar)

And also one proves in linear vector spaces that a vector(|>) combined with 0 by multiplicative law gives |0> .

Similarly can one prove that multiplication of a tensor of say rank 2,by 0 will produce a null rank 2 tensor.

 

[pessimist]

yes one can.

 

[tenchotomic]

yes one can.

Iam asking for an idea on how to do it?

 

[pmsrw3]

Iam asking for an idea on how to do it?

Well, in my books, a tensor is by definition a multilinear map from a bunch of vector spaces to the reals. The result you want comes right out of the linearity.

 

[CellCoree]

Yes, it is possible to prove that multiplication of a tensor of rank 2 by 0 will produce a null tensor of rank 2. This can be shown using the definition of tensor multiplication and the properties of null vectors in linear vector spaces.

First, let's define what it means for a tensor to be multiplied by 0. In tensor algebra, the multiplication of two tensors is defined as a linear combination of their components. Therefore, if one of the tensors is multiplied by 0, all of its components will become 0.

Now, let's consider a tensor of rank 2, represented by T_ij, and let's assume that it is multiplied by 0. This means that all of its components, T_ij, will become 0. In other words, the resulting tensor will have all of its components equal to 0, which is the definition of a null tensor.

Furthermore, we can also prove this using the properties of null vectors in linear vector spaces. In linear algebra, a null vector is defined as a vector that has all of its components equal to 0. Similarly, a null tensor of rank 2 can be defined as a tensor that has all of its components equal to 0. Therefore, when a tensor of rank 2 is multiplied by 0, it will produce a null tensor of rank 2, as all of its components will be equal to 0.

In conclusion, it is possible to prove that multiplication of a tensor of rank 2 by 0 will produce a null tensor of rank 2. This can be shown using the definition of tensor multiplication and the properties of null vectors in linear vector spaces.