What is the relationship between invertible linear mappings and rank in proofs?

In summary, if M is invertible, then the rank of the composite mapping M o L is equal to the rank of L, which is m. This is because M maps any k-dimensional subset of Rm onto a k-dimensional subset of Rm and the rank of M is equal to m.
  • #1
Mona1990
13
0
1. Hi!
I was wondering if anyone could help me to solve the following problem!
Let L : [R][n] ->[R][m] and M :[R][m]-> [R][m] be linear mappings.
Prove that if M is invertible, then rank (M o L) = rank (L)

thanks! :)
 
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  • #2
If M is invertible it maps Rm one-to-one onto Rm. In particular, it maps any k dimensional subset of Rm onto a k dimensional subset of Rm. Now, what does "rank" mean?
 
  • #3
the dimension of the column space of M is the rank of M

and we know that dim (null M)= 0 since the null space of M is just the zero vector
and since rank = m - (dimension of the null space of M)
so rank is m?
 

FAQ: What is the relationship between invertible linear mappings and rank in proofs?

What is a linear mapping?

A linear mapping, also known as a linear transformation, is a mathematical function that maps one vector space to another in a way that preserves the vector addition and scalar multiplication operations. In simpler terms, it is a function that takes in a vector and outputs another vector while maintaining the same structure and properties.

How is a linear mapping represented?

A linear mapping can be represented by a matrix. The columns of the matrix represent the basis vectors of the input space, while the rows represent the basis vectors of the output space. The entries in the matrix correspond to the coefficients of the linear combination of the basis vectors.

What is the difference between a linear mapping and a linear function?

While the terms are often used interchangeably, a linear mapping is a general concept that applies to vector spaces, while a linear function is a specific type of mapping that applies to real numbers. A linear function is a linear mapping between one-dimensional vector spaces.

How do you prove that a mapping is linear?

To prove that a mapping is linear, you must show that it satisfies two conditions: preservation of vector addition and preservation of scalar multiplication. This can be done by applying the properties of linear combinations and using the definition of a linear mapping.

Can a linear mapping have a negative determinant?

Yes, a linear mapping can have a negative determinant. The determinant of a linear mapping is a measure of how much the mapping distorts space. A negative determinant indicates that the mapping reverses the orientation of the space.

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