Matrix algebra : Find the matrix C such that N(A) = R(C)

In summary, the conversation revolves around clarifying the problem statement and equations related to finding the matrix C, which is a combination of X2 and X4. N(A) is the nullspace of A and R(C) is the rowspace of C. Further clarification is provided regarding the specific column (X4) and the mention of A and C in the problem.
  • #1
Arturo Andujo
3
0
Member advised to present information more clearly in future posts
Homework Statement
So I am wondering what the correct x[SUB]4[/SUB] is. From the method I was taught to expressing vector form I think that x[SUB]4[/SUB] should be the vector :
-3
0
4
1

But all of the examples I have come across translate into vector form directly after the RREF which would make it :
-3
4
0
1

Could someone clarify this for me?
Relevant Equations
Matrix A is the first matrix shown in my attached image
N(A) is the nullspace of A
R(C) is the range of C
20190325_204949.jpg
 
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  • #2
So what is the actual problem statement? Your relevant equations are merely a partial statement of the problem, but nowhere is it shown what A and C are. Presumbly N(A) means the nullspace of A. Is R(C) the rowspace of C?
 
  • #3
Mark44 said:
So what is the actual problem statement? Your relevant equations are merely a partial statement of the problem, but nowhere is it shown what A and C are. Presumbly N(A) means the nullspace of A. Is R(C) the rowspace of C?

Well my question pertains to X4 which is the 4th column. I wrote down what the Matrix A is in A=[]
I have to find C which will be X2 and X4 combined into a matrix.

N(A) is the nullspace of A
R(C) is the rowspace of C

Sorry I should have mentioned that from the start I hope this is enough clarification.
 

1. What is matrix algebra?

Matrix algebra is a branch of mathematics that deals with the manipulation and study of matrices, which are rectangular arrays of numbers or symbols. It involves operations such as addition, subtraction, multiplication, and division of matrices.

2. What is the meaning of N(A) = R(C) in matrix algebra?

In matrix algebra, N(A) refers to the null space of a matrix A, which is the set of all vectors that when multiplied by A result in the zero vector. R(C) refers to the range of a matrix C, which is the set of all possible outputs when C is multiplied by a vector. Therefore, N(A) = R(C) means that the null space of matrix A is equal to the range of matrix C.

3. How do you find the matrix C in N(A) = R(C)?

To find the matrix C in N(A) = R(C), you can use the following steps:

  1. Find the null space of matrix A by solving the system of linear equations Ax = 0.
  2. Choose a basis for the null space of A, and let it be represented by the columns of a matrix B.
  3. Find the transpose of matrix B, denoted as BT.
  4. The matrix C is then given by C = BTA.

4. What is the importance of finding the matrix C in N(A) = R(C)?

Finding the matrix C in N(A) = R(C) is important in solving systems of linear equations. It allows us to find a matrix that can transform the null space of matrix A into its range, making it easier to find solutions to the system. It also helps in understanding the relationship between the null space and range of a matrix.

5. Can the matrix C be unique in N(A) = R(C)?

No, the matrix C is not always unique in N(A) = R(C). This is because there can be multiple bases for the null space of matrix A, and each basis will result in a different matrix C. However, all these matrices will have the same null space and range, satisfying the equation N(A) = R(C).

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