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matqkks
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Why are we interested in looking at the kernel and range (image) of a linear transformation on a linear algebra course?
The kernel of a linear transformation is the set of all vectors in the domain that get mapped to the zero vector in the co-domain. In other words, it is the set of all inputs that produce an output of zero.
The kernel and the range of a linear transformation are complementary subspaces. This means that any vector in the range is perpendicular to any vector in the kernel, and vice versa. In addition, the dimension of the kernel and the dimension of the range add up to the dimension of the domain.
A linear transformation is injective if and only if the kernel contains only the zero vector. This means that the transformation maps each input to a unique output, as no two inputs can produce the same output.
Yes, the kernel can be empty if the only vector that gets mapped to the zero vector is the zero vector itself. This would mean that the transformation is injective and has a trivial kernel.
The kernel can be used to find the solutions to homogeneous systems of linear equations. This is because the solutions to such systems correspond to the vectors in the kernel of the coefficient matrix. By finding a basis for the kernel, we can find all the solutions to the system.