Finding x* in the Nullspace of A: What is its Significance?

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In summary, the nullspace of a matrix A is the set of all vectors x that satisfy the equation Ax = 0. Finding x* in the nullspace of A is significant because it helps determine the fundamental properties of A and understand the linear transformations it represents. To find x* in the nullspace of A, we can use row reduction techniques to solve the matrix equation Ax = 0. The sign of x* in the nullspace of A does not have any specific significance, as it represents coefficients of a linear combination. It is possible to find multiple solutions for x* in the nullspace of A due to its nature as a vector space.
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hoffmann
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Does anyone know how to approach this problem?

Let A be an m×n matrix of rank m, where m<n. Pick a point x in R^n, and let x∗ be the point in the nullspace of A closest to x. Write a formula for x∗ in terms of x and A.

What exactly is the significance of the point x* in the nullspace of A?
 
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i think this question has something to do with the orthogonality of subspaces and is analogous to saying the shortest distance between a point and an axis is just a perpendicular line...
 

What is the nullspace of a matrix?

The nullspace of a matrix A is the set of all vectors x that satisfy the equation Ax = 0. In other words, it is the set of all possible solutions for the matrix equation Ax = 0.

Why is finding x* in the nullspace of A significant?

Finding x* in the nullspace of A is significant because it allows us to determine the fundamental properties of the matrix A. It helps us understand the relationship between the columns and rows of A and provides insight into the linear transformations represented by A.

How do we find x* in the nullspace of A?

To find x* in the nullspace of A, we can use the techniques of row reduction to solve the matrix equation Ax = 0. This will give us the values of x that satisfy the equation and thus, belong in the nullspace of A.

What does the sign of x* in the nullspace of A represent?

The sign of x* in the nullspace of A does not have any specific significance. The values of x* represent the coefficients of the linear combination that produces the zero vector, and the sign of these coefficients can vary depending on the specific values of A.

Can we find multiple solutions for x* in the nullspace of A?

Yes, it is possible to find multiple solutions for x* in the nullspace of A. This is because the nullspace of A is a vector space, and any linear combination of vectors in the nullspace will also be in the nullspace. Therefore, there can be an infinite number of solutions for x* in the nullspace of A.

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