# Check of a problem about nullspace

• Zero2Infinity
In summary, the problem involves finding a matrix A associated to a linear map f, such that its nullspace is the subspace V generated by (1,1,0) and (0,2,0) in R^3. The solution involves using the nullspace definition and the standard basis of R^3, resulting in a matrix A = [0 0 1; 0 0 1; 0 0 1]. However, it is unclear if W, defined as the set of all (x,y,z) in R^3 such that x-y=0, is relevant to the problem. If it is, then the solution may not be correct.
Zero2Infinity

## Homework Statement

Let ##V\subset \mathbb{R}^3## be the subspace generated by ##\{(1,1,0),(0,2,0)\}## and ##W=\{(x,y,z)\in\mathbb{R}^3|x-y=0\}##. Find a matrix ##A## associated to a linear map ##f:\mathbb{R}^3\rightarrow\mathbb{R}^3## through the standard basis such that its nullspace is ##V##.

## Homework Equations

Nullspace definition

## The Attempt at a Solution

Using the nullspace definition I get that ##f(0,2,0)=f(1,1,0)=(0,0,0)##. Thus,
$$f(0,2,0) =(0,0,0)$$
$$f(1,1,0) = (0,0,0)$$
$$f(0,0,1)=(1,1,1)$$
Since the matrix has to be written wrt the standard basis of ##\mathbb{R}^3##, which is ##(1,0,0),(0,1,0),(0,0,1)##, I infer that
$$f(1,0,0)=f(1,1,0)-\frac{1}{2}f(0,2,0)=(0,0,0)-\frac{1}{2}(0,0,0)=(0,0,0)$$
$$f(0,1,0)=\frac{1}{2}f(0,2,0)=\frac{1}{2}(0,0,0)=(0,0,0)$$
while I know from the text of the problem that ##f(0,0,1)=(1,1,1)##. In conclusion,
$$A=\begin{pmatrix} 0&0&1\\0&0&1\\0&0&1\end{pmatrix}$$
If I look for a basis of the kernel I effectively get the two vectors ##(1,1,0),(0,2,0)##, so it should be ok. Is it correct?

Zero2Infinity said:
Let ##V\subset \mathbb{R}^3## be the subspace generated by ##\{(1,1,0),(0,2,0)\}## and ##W=\{(x,y,z)\in\mathbb{R}^3|x-y=0\}##.
Does this mean
1. Let ##V\subset \mathbb{R}^3## be the subspace generated by ##\{(1,1,0),(0,2,0)\}## and let ##W=\{(x,y,z)\in\mathbb{R}^3|x-y=0\}##; or does it mean
2. Let ##V\subset \mathbb{R}^3## be the subspace generated by ##\{(1,1,0),(0,2,0)\}\cup\{(x,y,z)\in\mathbb{R}^3|x-y=0\}##?
If it means the second one then ##V=\mathbb R^3##, so ##f## must be the zero map. If it means the first one then ##W## is not used in the problem, which raises the question of why it has been defined.

## 1. What is the nullspace of a matrix?

The nullspace of a matrix is the set of all vectors that, when multiplied by the matrix, result in a zero vector. In other words, it is the set of solutions to the equation Ax = 0, where A is the given matrix and x is a vector of unknowns.

## 2. How do you check the nullspace of a matrix?

To check the nullspace of a matrix, you can use the row reduction method to reduce the matrix to its reduced row echelon form (RREF). The columns corresponding to the pivot positions in the RREF will form the basis for the nullspace.

## 3. What is the dimension of the nullspace?

The dimension of the nullspace is equal to the number of free variables in the RREF of the matrix. This can also be interpreted as the number of columns in the matrix minus the rank of the matrix.

## 4. Why is the nullspace important?

The nullspace is important because it provides valuable information about the solutions to a system of linear equations. It can also be used to determine if a system of equations has a unique solution, no solution, or infinitely many solutions.

## 5. How is the nullspace related to the eigenvalues of a matrix?

The nullspace is closely related to the eigenvalues of a matrix. The eigenvalues of a matrix A are the values λ for which the equation Ax = λx has a non-trivial solution. This solution will be in the nullspace of A-λI, where I is the identity matrix. Therefore, the nullspace plays a key role in determining the eigenvalues of a matrix.

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