- #1
sarumman
- 2
- 1
Homework Statement
given
Homework Equations
rank
dim
null space
The Attempt at a Solution
I tried to base my answer based on the fact that null A and null A^2 is Contained in F (n)
and
dim N(A)+rank(A)=N
same goes for A^2.
The null space of a matrix, also known as the kernel, is the set of all vectors that when multiplied by the matrix, result in a vector of zeros. In other words, it is the set of all solutions to the homogeneous equation Ax = 0, where A is the given matrix.
To find the null space of a matrix, you can use Gaussian elimination or row reduction to put the matrix in reduced row echelon form. The columns without pivot positions correspond to the free variables, and the null space can be expressed as a linear combination of these variables.
The dimension of the null space of a matrix is also known as the nullity, and it is equal to the number of free variables in the reduced row echelon form of the matrix. In other words, it is the number of linearly independent vectors in the null space.
The null space and column space of a matrix are complementary subspaces. This means that the dimension of the null space and the dimension of the column space add up to the total number of columns in the matrix.
The null space is significant in linear algebra as it helps us understand the solutions to systems of linear equations. It can also be used to find the basis for the column space of a matrix, which is useful in solving systems of equations and finding solutions to linear transformations.