Question

what is the Kernel and the Range of a linear transformation?

Answer 1

The kernel is the set of verctors in V that maps onto \(\vec{0}\)

Answer 2

Basically the same thing as nullspace.

Answer 3

The range is the column space.

Answer 4

Let \(T:V\rightarrow W\) be a linear transformation. Then the set of all vectors \(\vec{v}\) in \(V\) that satisfy \(T(\vec{v})=\vec{0}\) is called the kernel of \(T\).

Answer 5

Ok I got the Kernel but what is the column space?

Answer 6

Do an RREF on your transform matrix. Some columns will be identity based, some will be free variables, right? In the for the columns in the ORIGINAL matrix that have the identity in the RREF matrix, that is the column space of the matrix.

Answer 7

\[\left[\begin{matrix}1 & 1 & -1 \\1 & 2 & -3\\ 1 & 0 & 1 \\1 & 3 & -5\end{matrix}\right]\rightarrow \mathrm{RREF}\rightarrow
\left[\begin{matrix}1 & 0 & 1 \\0 & 1 & 2\\ 0 & 0 & 0 \\0 & 0 & 0\end{matrix}\right]\]
Because the reduced row echelon form on the right has pivots or 1s in the first two columns, those columns in the original are the range of it.
\[\mathrm{range}(T)=\mathrm{span}\left\{ \left[\begin{matrix}1\\1\\1\\1\end{matrix}\right],\left[\begin{matrix}1\\2\\0\\3\end{matrix}\right]\right\}\]

Answer 8

do you know what the helper mean? If you have a specific problem, post it , after 2 or 3 times practicing, you will understand it. Kernel concept looks easy but it's not easier than rank, column space or null space or else. You can understand kernel, then there is no reason to stuck at column space.

Answer 9

thanx