Linear maps (rank and nullity)

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In summary, the rank-nullity theorem states that for a linear map T: ℝ^{m}-> R^{n} with rank k, the nullity of T is equal to m-k. The number of columns for the map is determined by the dimension of the starting space, which is m. Therefore, the nullity of T can be calculated as m-k.
  • #1
NewtonianAlch
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Homework Statement


A linear map T: ℝ[itex]^{m}[/itex]-> R[itex]^{n}[/itex] has rank k. State the value of the nullity of T.

The Attempt at a Solution



I know that rank would be the number of leading columns in a reduced form and the nullity would simply be the number of non-leading columns, or total columns - rank.

As we are not given how many different vectors are in the matrix, I'm not too sure how to give a value.
 
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  • #2
NewtonianAlch said:
nullity would simply be [...] total columns - rank.

The rank is given.
What is the total number of columns for a map [itex]\mathbb{R}^m \to \mathbb{R}^n[/itex]?
 
  • #3
I'm inclined to say m, but I'm not entirely sure. So m - k? Considering that's the starting space if you will.
 
  • #4
The "rank-nullity theorem" which, if you were given this problem, you are probably expected to know, says that if A is a linear transformation from a vector space of degree m to a vectoer space of degree n then rank(A)+ nullity(A)= m.
 
  • #5
NewtonianAlch said:
I'm inclined to say m, but I'm not entirely sure. So m - k? Considering that's the starting space if you will.

If it's a map from [itex]\mathbb{R}^m[/itex] to [itex]\mathbb{R}^n[/itex], that means that you should be able to multiply the matrix with a vector with m components, and this should give a vector with n components. If you think about the way matrix multiplication works, does the number of columns need to be m or n?
 

Related to Linear maps (rank and nullity)

1. What is the definition of rank and nullity in linear maps?

Rank and nullity are two important characteristics of linear maps, which are functions that preserve linear combinations between vector spaces. The rank of a linear map is the dimension of its image, or the number of linearly independent vectors in the output. The nullity of a linear map is the dimension of its kernel, or the number of linearly independent vectors in the input that are mapped to the zero vector.

2. How do you calculate the rank and nullity of a linear map?

To calculate the rank of a linear map, you can find a basis for its image and count the number of vectors in the basis. To calculate the nullity, you can find a basis for its kernel and count the number of vectors in the basis. Alternatively, you can use the rank-nullity theorem, which states that the sum of the rank and nullity of a linear map is equal to the dimension of its domain.

3. What is the relationship between the rank and nullity of a linear map?

The rank and nullity of a linear map are complementary, meaning that they add up to the dimension of the domain. This is because the dimension of the image and the dimension of the kernel must add up to the dimension of the domain, according to the rank-nullity theorem. In other words, the more vectors that are mapped to the zero vector (higher nullity), the fewer vectors that are mapped to linearly independent outputs (lower rank).

4. How can the rank and nullity of a linear map be used in solving systems of linear equations?

The rank and nullity of a linear map can be used to determine whether a system of linear equations has a unique solution, infinite solutions, or no solutions. If the rank of the coefficient matrix is equal to the number of variables, then the system has a unique solution. If the rank is less than the number of variables, then the system has infinite solutions. If the rank is greater than the number of variables, then the system has no solutions.

5. What is the significance of the rank and nullity of a linear map in linear algebra?

The rank and nullity of a linear map provide important information about its structure and behavior. They can be used to determine whether a linear map is one-to-one or onto, and to characterize its range and null space. They also have applications in fields such as differential equations, optimization, and data analysis. Additionally, the rank-nullity theorem is a fundamental result in linear algebra that relates the dimensions of different vector spaces.

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