Nullspace Matrix Homework: Struggling to Find an Example Vector

In summary: Try entering (4;0,6;3) or (4;-1,6;3). Hey!In summary, the portal is telling me that my vectors are wrong and I need to try a different basis.
  • #1
negation
818
0

Homework Statement



Let A be the matrix:

[3,3,-2,0;-3,-3,3,-2]


a) An example of a vector in the nullspace of A is
b) An example of a vector NOT in the nullspace of A is


Sorry guy but I'm really STRUGGLING

The Attempt at a Solution




a) I found x1 ,x2,x3,x4 = -x2+4/3x4, x2, 2x4, x4

=x2( -1,1,0,0) + x4(4/3,0,2,1)

For more than an hour, the answer sheet is telling that my vectors are wrong.
I don't see how it can be wrong.

Ax = 0
I reduced A to row echelon form, then solve in terms of the free variables x2 and x4.
 
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  • #2
That answer is correct as you can see by apply A to each of (-1,-, 0, 0) and (4/3, 0, 2, 1). The nullspace is two dimensional and there are an infinite number of bases for it. How is the answer sheet telling you that you are wrong? If it gives a different pair of vectors, that may just be a different basis for the same space. For example, if (4/3, 0, 2, 1) is a basis vector, so is (4, 0, 6, 3).
 
  • #3
HallsofIvy said:
That answer is correct as you can see by apply A to each of (-1,-, 0, 0) and (4/3, 0, 2, 1). The nullspace is two dimensional and there are an infinite number of bases for it. How is the answer sheet telling you that you are wrong? If it gives a different pair of vectors, that may just be a different basis for the same space. For example, if (4/3, 0, 2, 1) is a basis vector, so is (4, 0, 6, 3).

The answer sheet is an online portal prompting me of my error whenever I submit the answer.
I know my answer has to be correct. I submit ( -1,1,0,0),(4/3,0,2,1) in vector form. and I know my answer cannot be wrong because 1) I perform RREF on the matrix A then, 2) express the solution as free variables.

I do not know what is going on with the portal prompting me of my error.

edit: Message: "What you have given as vector in the nullspace of A is not of the correct dimension."
 
  • #4
I'll solve part (b) first:

so, a vector that is not in the nullspace implies ax =/= 0

[3,3,-2,0;-3,-3,3,-2] [x1,x2,x3,x4] = [0;1]

I get

x1 = 2/3 - x2 + 4/3x4
x3 = 1 + 2x4
free : x2 , x4

x1,x2,x3,x4 = ( 2/3 - x2 + 4/3x4, x2, 1+2x4, x4)
 
  • #5
A has four columns so acts on four dimensional vectors. If you gave (-1, 1, 0, 0) and (4/3, 0, 2, 1) they have the correct dimension. How did you enter the fraction 4/3? The answer "sheet" might be interpreting that as (4, 3, 0, 2, 1). Try (4, 0, 6, 3), as I suggested before, instead.
 
  • #6
HallsofIvy said:
A has four columns so acts on four dimensional vectors. If you gave (-1, 1, 0, 0) and (4/3, 0, 2, 1) they have the correct dimension. How did you enter the fraction 4/3? The answer "sheet" might be interpreting that as (4, 3, 0, 2, 1). Try (4, 0, 6, 3), as I suggested before, instead.

I entered the fraction as it is, that is- 4/3. There is a button that shows my answer in some kind of tex format and it corresponds to 4/3 as how one would normally write a value as a fraction form.


Untitled.png
 
  • #7
HallsofIvy said:
A has four columns so acts on four dimensional vectors. If you gave (-1, 1, 0, 0) and (4/3, 0, 2, 1) they have the correct dimension. How did you enter the fraction 4/3? The answer "sheet" might be interpreting that as (4, 3, 0, 2, 1). Try (4, 0, 6, 3), as I suggested before, instead.

It isn't working. There isn't any reason for me to enter (4,0,..) because the system does in fact recognize 4/3 as an input language for the answer.
 
  • #8
negation said:
It isn't working. There isn't any reason for me to enter (4,0,..) because the system does in fact recognize 4/3 as an input language for the answer.

Hey!

I might be wrong, but the way you are inputting the answer might be incorrect.

Notice the vectors you are putting in are 1x4, and you know you should be inputting 4x1.

Maybe try putting a semicolon after your numbers instead?

If that doesn't work.. well, maybe there is an error in the program.
 

Related to Nullspace Matrix Homework: Struggling to Find an Example Vector

1. What is a nullspace matrix?

A nullspace matrix, also known as a kernel matrix, is a square matrix that represents the linear transformations between vector spaces. It is created by setting the determinant of the matrix to zero and solving for the variables.

2. Why is finding an example vector for a nullspace matrix important?

Finding an example vector for a nullspace matrix is important because it helps to understand the linear transformations that the matrix represents. It also provides insight into the solutions of a system of linear equations represented by the matrix.

3. What makes finding an example vector for a nullspace matrix difficult?

The difficulty in finding an example vector for a nullspace matrix lies in the fact that there are infinite possible solutions. It requires a deep understanding of linear algebra and the ability to manipulate equations and variables to find a suitable vector.

4. What are some strategies for finding an example vector for a nullspace matrix?

One strategy is to start by setting one or more variables to arbitrary values and then solving for the other variables. Another strategy is to use row reduction to transform the matrix into an echelon form and then use the resulting equations to find a suitable vector.

5. How can I improve my skills in finding example vectors for nullspace matrices?

To improve your skills in finding example vectors for nullspace matrices, it is important to practice solving various types of systems of linear equations. You can also study the properties of nullspace matrices and learn different strategies for finding example vectors. Seeking help from a tutor or joining a study group can also be beneficial.

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