When is the kernel 0

Related to linear transformations is the idea of the kernel of a linear transformation. The kernel of a linear transformation L is the set of all vectors v such that. Let L be the linear transformation from M 2x2 to P 1 defined by. Then to find the kernel of L , we set.

The kernel of a linear transformation from a vector space V to a vector space W is a subspace of V. Suppose that u and v are vectors in the kernel of L. We can conclude that the kernel of L is a subspace of V. In light of the above theorem, it makes sense to ask for a basis for the kernel of a linear transformation. In the previous example, a basis for the kernel is given by. Next we show the relationship between linear transformations and the kernel.

Let L be and let v be in Ker L. We need to show that v is the zero vector. We have both. Since L is ,. We have seen that a linear transformation from V to W defines a special subspace of V called the kernel of L. Now we turn to a special subspace of W. Let L be a linear transformation from a vector space V to a vector space W.

Then the range of L is the set of all vectors w in W such that there is a v in V with. The range of a linear transformation L from V to W is a subspace of W. Let w 1 and w 2 vectors in the range of W. Then there are vectors v 1 and v 2 with. Hence the range of L is a subspace of W. We say that a linear transformation is onto W if the range of L is equal to W. Let L be the linear transformation from R 2 to R 3 defined by. Find a basis for Ker L.

Determine of L is Find a basis for the range of L. Determine if L is onto. The Ker L is the same as the null space of the matrix A. We have. Hence a basis for Ker L is. L is not since the Ker L is not the zero subspace. Now for the range. These two vectors are just the columns of A. In general. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. This is by definition. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. If the kernel of a matrix is the 0 vector, why is the basis of the kernel non existent?

Asked 5 years, 8 months ago. Active 4 years, 10 months ago. Viewed 13k times. Brad Brad 11 1 1 gold badge 1 1 silver badge 3 3 bronze badges. I know that the basis exists.