How to Express Null(A) in Cartesian Form?

In summary, the conversation is discussing how to express the null space of A in cartesian form. The solution involves finding the one dimensional vector space through the vector (-2, 1, 1) and expressing it as ( -2; 1; 1)*x3. It is unclear if this is sufficient or if it should be expressed as the line -x = 2y = 2z.
  • #1
canyonist
2
0
Hi there,

just a pretty straight forward query I need cleared up...

If a question asks for the null space of A in cartesian form how do I set it out?

This is what I've got:

X = ( -2; 1; 1)*x3 for all values of x3

Therefore, Null(A) corresponds geometrically to the line through the origin and (-2, 1, 1).

Now do I need to say that this is equivelant to -x = 2y = 2z for it to be in cartesian form or will the above suffice?

thanks for the help
 
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  • #2
I'm a bit confused. What is A, and how does that compare to X? Is X supposed to be the one dimensional vector space through the vector (-2,1,1)?
 
  • #3
Yes, so AX = 0


where X = ( x1; x2; x3) and A = (1 0 2; 0 3 -3; 4 2 6)

which can be expressed as ( -2; 1; 1)*x3 after row reduction and back substitution.

I need to know whether I can leave it at that, or express it as the line: -x = 2y = 2z
 

What is Null(A)?

Null(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. In other words, it is the set of all solutions to the homogeneous equation Ax = 0.

What is the significance of Null(A)?

The null space of a matrix A is important because it provides information about the solutions to the equation Ax = b, where b is a vector. If b is not in the null space, then the equation has no solution. If b is in the null space, then there are an infinite number of solutions.

How is Null(A) related to the rank of a matrix?

The rank of a matrix is equal to the number of linearly independent columns in the matrix. The dimension of the null space, also known as the nullity, is equal to the number of linearly dependent columns. Therefore, the rank plus the nullity of a matrix equals the number of columns in the matrix.

What is cartesian form of a matrix?

Cartesian form of a matrix is a way of representing a matrix as a combination of its column vectors. It is also known as the column space of a matrix. It is written as a linear combination of the column vectors with coefficients multiplied to each vector. For example, if A is a 3x3 matrix, its cartesian form would be written as x1A1 + x2A2 + x3A3, where x1, x2, and x3 are coefficients and A1, A2, and A3 are the column vectors.

How are Null(A) and cartesian form related?

The dimension of the null space of a matrix A is equal to the number of linearly dependent columns, which is also equal to the number of columns minus the rank of the matrix. The linearly dependent columns can be removed from the cartesian form, resulting in a simplified form of the matrix. Therefore, the null space and cartesian form are related because they both provide information about the linear independence of the columns in a matrix.

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