What does it mean when Ax B has a solution?

Ax = b has a solution if and only if b is a linear combination of the columns of A. Note: If A does not have a pivot in every row, that does not mean that Ax = b does not have a solution for some given vector b. It just means that there are some vectors b for which Ax = b does not have a solution.

How do you find the general solution of Ax B?

One way to find a particular solution to the equation Ax = b is to set all free variables to zero, then solve for the pivot variables. The general solution to Ax = b is given by xcomplete = xp + xn, where xn is a generic vector in the nullspace.

What is the kernel of a transformation?

The kernel or null-space of a linear transformation is the set of all the vectors of the input space that are mapped under the linear transformation to the null vector of the output space.

What does kernel mean in linear algebra?

In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector.

What is the difference between Ax B and ax 0?

Ax = 0 is a homogeneous equations and Ax = b = 0 is a nonhomogeneous equation.

How many solutions does the equation Ax B have explain your answer?

The equation Ax = b has an infinite number of solutions. The equation Ax = 0 has a non-zero solution.

What is Ax B when does Ax B has a unique solution?

The system AX = B has a unique solution provided dim(N(A)) = 0. Since, by the rank theorem, rank(A) + dim(N(A)) = n (recall that n is the number of columns of A), the system AX = B has a unique solution if and only if rank(A) = n.

What is a SVM kernel?

A kernel is a function used in SVM for helping to solve problems. They provide shortcuts to avoid complex calculations. The amazing thing about kernel is that we can go to higher dimensions and perform smooth calculations with the help of it. We can go up to an infinite number of dimensions using kernels.

What is a kernel in math?

From Wikipedia, the free encyclopedia. In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1).

What is the relationship between the kernel of a linear transformation from a finite dimensional vector space V to a vector space W to the nulL space of a matrix?

The kernel of a linear transformation from a vector space V to a vector space W is a subspace of V. Hence u + v and cu are in the kernel of L. We can conclude that the kernel of L is a subspace of V.

How to find the kernel of a matrix with Ax = 0?

To find the kernel of a matrix A is the same as to solve the system AX = 0, and one usually does this by putting A in rref. The matrix A and its rref B have exactly the same kernel.

How to solve ax = b with a particular solution?

A particular solution One way to find a particular solution to the equation Ax = b is to set all free variables to zero, then solve for the pivot variables. For our example matrix A, we let x2 = x4 = 0 to get the system of equa­ tions: x1 + 2×3 = 1 2×3 = 3 1. which has the solution x3 = 3/2, x1 = −2.

How do you find the kernel of a linear equation?

In both cases, the kernel is the set of solutions of the corresponding homogeneous linear equations, AX = 0 or BX = 0. You can express the solution set as a linear combination of certain constant vectors in which the coefficients are the free variables. E.g., to get the kernel of and then one solves x+2y+3z = 0 (this is already reduced).

Why are the vectors produced to span the kernel always independent?

span the kernel, clearly. They are independent because, each one, in the coordinate spot corresponding to the free variable which is its coefficient, has a 1, while the other vector(s) have a 0 in that spot. So the vectors produced to span the kernel by this method are always a basis for…