charlies1902 said:Let's say you have a 3x3 matrix and it's invertible. Let's call it A
If you were to find a basis for the nullspace of A, would the basis just be the original 3 column vectors of A?
charlies1902 said:It would be the column vectors of A right?
Dick said:Ok, so you don't know what "invertible" means. Could you maybe look it up?
charlies1902 said:det≠0 and a pivot is in every column for RREF(A).
So for a 3x3 invertible matrix,it's basis is [1 0 0]^t [0 1 0]^t and [0 0 1]^t?
Dick said:That's an example of an invertible matrix. What vectors are in its null space?
charlies1902 said:The 0 vector?
Dick said:Yes. Wouldn't that always be the only answer if A were invertible?
charlies1902 said:I think I got it confused with the column space.
A basis for the column space for this case would be the original 3 column vectors if A right?
The nullspace of an invertible 3x3 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 all solutions to the equation Ax = 0, where A is the 3x3 matrix.
To find the nullspace of an invertible 3x3 matrix, you can use the method of Gaussian elimination to reduce the matrix to reduced row echelon form. The columns corresponding to the pivots in the reduced matrix will form the basis for the nullspace.
The dimension of the nullspace of an invertible 3x3 matrix is equal to the number of non-pivot columns in the reduced matrix. In other words, it is equal to the number of free variables in the system of equations Ax = 0.
No, the nullspace of an invertible 3x3 matrix cannot be empty. This is because an invertible matrix always has three pivot columns, which means there will always be at least one non-pivot column and therefore at least one vector in the nullspace.
The nullspace of an invertible 3x3 matrix can help us understand the linear transformations represented by the matrix. It can also provide insights into the solutions of systems of equations and the properties of the matrix, such as rank and eigenvalues.