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cragar
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Can someone give me a layman’s terms explanation of what a null space is .
Null space is a mathematical concept that refers to the set of all possible solutions to a system of equations where the output (or dependent variable) is equal to zero. In other words, it is the set of inputs (or independent variables) that do not affect the output.
Understanding null space is important in various fields of science and engineering, such as linear algebra, physics, and control systems. It allows us to solve systems of equations and determine the relationship between inputs and outputs in a system.
Null space is closely related to the concept of linear independence. If a set of vectors in a vector space are linearly independent, then the null space of those vectors will only contain the zero vector. In other words, there are no other solutions to the system of equations other than the trivial solution (where all variables are equal to zero).
Yes, null space can be visualized in certain cases. In a two-dimensional space, the null space of a vector can be represented by a line passing through the origin. In a three-dimensional space, the null space of a vector can be represented by a plane passing through the origin. However, in higher dimensions, it becomes more difficult to visualize null space.
The rank-nullity theorem states that the sum of the rank (the number of linearly independent rows or columns) and the nullity (the dimension of the null space) of a matrix is equal to the number of columns. In other words, it shows the relationship between the dimensions of the null space and the range (or column space) of a matrix.