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What does the null space of a matrix A represent?
According to the Rank-Nullity Theorem, the dimension of the null space is equal to n minus the rank of A.
True
What is the mathematical expression representing the relationship between the number of variables n, the rank of A, and the dimension of the null space of A?
n = r(A) + dim(Null(A))
If every vector v is a solution to Ax = 0, then A is the _____ matrix.
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Which statement is true regarding the Rank-Nullity Theorem?
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How can k variables affect the dimension of the null space?
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Match the following statements with their corresponding concepts:
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The dimension of the null space can also be described as the number of _____ of the solution.
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Study Notes
Null Space and Rank-Nullity Theorem
- Definition: The null space of a matrix A is the set of all vectors x that satisfy the homogeneous equation Ax = 0.
- Rank-Nullity Theorem: For a homogeneous system of equations with n variables, the dimension of the null space (the set of solutions) is equal to n minus the rank of A.
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Proof:
- The theorem states that the dimension of the null space is n - r(A), which is the number of degrees of freedom in the solution.
- We can choose k variables arbitrarily, where k = n - r(A), and the remaining variables are expressed as linear combinations of those k variables.
- This creates a set of k linearly independent vectors that span the null space, making it a basis for the null space.
- Therefore, the dimension of the null space is k, which is n - r(A).
Corollary of Rank-Nullity Theorem
- Statement: For a homogeneous system Ax = 0, the number of variables n is equal to the sum of the rank of A and the dimension of the null space of A.
- Mathematical Expression: n = r(A) + dim(Null(A))
- Meaning: The number of variables in a homogeneous system is equal to the rank of the coefficient matrix plus the dimension of its null space.
Other Theorems Related to Null Space
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Theorem 1: If Av = 0 for every vector v in F^n, then A is the zero matrix.
- Proof: If every vector is a solution to Ax = 0, the null space is F^n and has dimension n.
- By the Rank-Nullity theorem, r(A) = 0.
- A matrix with rank zero must be the zero matrix.
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Theorem 2: If Av = Bv for every vector v, then A = B (where A and B are both m x n matrices).
- Proof: The equation Av = Bv can be rewritten as (A - B)v = 0 for every v.
- By Theorem 1, A - B is the zero matrix, therefore A = B.
Future Topics
- The concepts of null spaces and the Rank-Nullity Theorem become even more useful when discussing linear transformations.
- Later discussions will include determinants and invertible matrices.
Null Space
- The null space of a matrix A is the set of all vectors x that satisfy the equation Ax = 0.
- These vectors are considered solutions to the homogeneous equation.
Rank-Nullity Theorem
- For a homogeneous system of equations with n variables, the dimension of the null space is equal to n minus the rank of A (denoted as r(A)).
- This theorem can be expressed as: dim(Null(A)) = n - r(A).
- The dimension of the null space represents the number of degrees of freedom in the solution.
- By choosing k variables freely, where k = n - r(A), the remaining variables can be determined as linear combinations of those k variables.
- This results in a set of k linearly independent vectors that span the null space, forming a basis for the space.
Corollary of Rank-Nullity Theorem
- For a homogeneous system Ax = 0, the number of variables n is equal to the sum of the rank of A and the dimension of the null space of A.
- This can be expressed as: n = r(A) + dim(Null(A)).
Other Theorems Related to Null Space
- If Av = 0 for every vector v in F^n, then A is the zero matrix.
- This is because if every vector is a solution to Ax = 0, the null space is F^n and has dimension n.
- By the Rank-Nullity theorem, r(A) = 0, which results in A being the zero matrix.
- If Av = Bv for every vector v in F^n (where A and B are both m x n matrices), then A = B.
- This can be shown by rewriting the equation as (A - B)v = 0 for every v.
- According to the previous theorem, A - B is the zero matrix, therefore A = B.
Future Topics
- The concepts of null spaces and the Rank-Nullity Theorem are crucial in understanding linear transformations.
- Later discussions will explore determinants and invertible matrices.
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Description
Explore the concepts of null space and the Rank-Nullity Theorem in linear algebra. This quiz will test your understanding of how the dimension of the null space relates to the rank of a matrix and the implications for solving homogeneous systems of equations.
