thanks Let V and W be finite-dimensional vector spaces. The nullspace of a linea

Question

thanks

Let V and W be finite-dimensional vector spaces. The nullspace of a linear transformation L: V rightarrow W is the set of X V such that L(X) = 0. It was shown in Exercise 28 on page 167 that the nullspace is a subspace of V. Let ordered bases B and B for V and W be given and let M be the matrix of L with respect to these bases. Show that L(X) = 0 if and only if MTCBX = 0. Let T: V right arrow W be a linear transformation between two vector spaces. We define the nullspace of T to be the set of X such that T (X) = 0. Show that the nullspace of T is a subspace of V.

Explanation / Answer

Since C is not defined, I'll be more generic.

Using the formula of change of basis between the canonical basis of V,W , we can write :

MP = QM2 where P,Q are the corresponding change of basis and M2 is the matrix of L in the canonical basis of V and W.

L(X)=0 <=> M2X=0 <=> QM2X=0 (since Q is invertible ) <=> MPX=0.

I let you fix the name of the basis and matrix, basically you just have to use the change of basis formula.

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