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could someone kinda explain in plain english what null space is?
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0rthodontist said:Also, geometrically, if you're dealing with matrices, then the null space of A is the set of all vectors perpendicular to each vector in the row space of A.
Null space, also known as the kernel, is the set of all vectors that are mapped to the zero vector by a given linear transformation. In simpler terms, it is the set of all solutions or inputs that result in an output of zero.
Understanding null space is important because it helps us to understand the behavior of a linear transformation. It also allows us to determine the dimension of the vector space, and provides insight into the types of solutions a system of linear equations may have.
Null space and linear independence are closely related because the null space contains all of the linearly dependent vectors in a vector space. In other words, a vector is in the null space if and only if it can be written as a linear combination of other vectors in the space.
Yes, it is possible for null space to be empty. This occurs when a linear transformation maps all vectors in the vector space to a non-zero vector, meaning there are no inputs that result in an output of zero. In this case, the null space is said to be trivial.
The null space of a matrix can be found by solving the homogeneous system of linear equations represented by the matrix. This can be done using techniques such as Gaussian elimination or finding the reduced row echelon form. The solutions to this system will give the basis for the null space.