Question on rank-nullity theorem

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In summary, the basis (0, 1, -1) is chosen because it satisfies the condition for the kernel of g and all vectors satisfying this condition are scalar multiples of (0, 1, -1). Therefore, (0, 1, -1) forms a basis for the kernel of g and the dimension of the kernel is 1.
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SarahER
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So my question on this is: Why is the basis (0, 1, -1) because the dim is 1?
 

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SarahER said:
View attachment 3493

So my question on this is: Why is the basis (0, 1, -1) because the dim is 1?
Hi Sarah, and welcome to MHB!

If the vector $(a,b,c)$ satisfies $a=0$ and $b+c=0$, then $c=-b$ and the vector is equal to $(0,b,-b) = b(0,1,-1).$ So all such vectors are scalar multiples of $(0,1,-1)$. Thus the single vector $(0,1,-1)$ forms a basis for the kernel of $g$, which therefore has dimension 1.
 
  • #3
Opalg said:
Hi Sarah, and welcome to MHB!

If the vector $(a,b,c)$ satisfies $a=0$ and $b+c=0$, then $c=-b$ and the vector is equal to $(0,b,-b) = b(0,1,-1).$ So all such vectors are scalar multiples of $(0,1,-1)$. Thus the single vector $(0,1,-1)$ forms a basis for the kernel of $g$, which therefore has dimension 1.

So, is it 1 dimensional because b = -c?
 

FAQ: Question on rank-nullity theorem

1. What is the rank-nullity theorem?

The rank-nullity theorem, also known as the fundamental theorem of linear algebra, states that for a given vector space V, the sum of the dimensions of the null space and the column space is equal to the dimension of V. In other words, the number of linearly independent columns plus the number of linearly dependent columns is equal to the total number of columns in a matrix.

2. Why is the rank-nullity theorem important?

The rank-nullity theorem is important because it provides a way to understand the relationships between the dimensions of a vector space and its subspaces. It also helps to determine the number of free variables in a system of linear equations, which is crucial for solving them.

3. Can the rank-nullity theorem be applied to any vector space?

Yes, the rank-nullity theorem applies to any finite-dimensional vector space. It can also be extended to infinite-dimensional vector spaces, but with some modifications.

4. How is the rank-nullity theorem used in real-world applications?

The rank-nullity theorem has many applications in fields such as computer science, engineering, and physics. It is used to solve systems of linear equations, analyze data in statistics, and design algorithms for data compression and image processing.

5. Are there any limitations to the rank-nullity theorem?

While the rank-nullity theorem is a powerful tool in linear algebra, it does have some limitations. It only applies to finite-dimensional vector spaces and cannot be used for infinite-dimensional vector spaces. It also assumes that all vectors are real or complex numbers, and does not apply to other types of vector spaces such as matrices or polynomials.

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