Finding matrix using linear transformations

**chadlyter** · June 23rd 2008, 07:07 AM

Let B = (v1, v2, v3, v4) be a basis for a vector space V. Find the matrix with respect to B of the linear operator T: V -----> V defined by:

T(v1) = (v2), T(v2) = v3, T(v3) = v4, T(v4) = v1

**TheEmptySet** · June 23rd 2008, 07:47 AM

Originally Posted by chadlyter

Let B = (v1, v2, v3, v4) be a basis for a vector space V. Find the matrix with respect to B of the linear operator T: V -----> V defined by:

T(v1) = (v2), T(v2) = v3, T(v3) = v4, T(v4) = v1

When you are transforming basis $e_i$ vectors the matrix will be

$T=\begin{bmatrix}T(e_1) && T(e_2) && T(e_3) && T(e_4) \end{bmatrix}$

Or in your case

$T=<br /> \begin{bmatrix}T(v_1) && T(v_2) && T(v_3) && T(v_4) \end{bmatrix}=<br /> \begin{bmatrix}v_2 && v_3 && v_4 && v_1 \end{bmatrix}$

**Opalg** · June 23rd 2008, 08:07 AM

Originally Posted by chadlyter

Let B = (v1, v2, v3, v4) be a basis for a vector space V. Find the matrix with respect to B of the linear operator T: V -----> V defined by:

T(v1) = (v2), T(v2) = v3, T(v3) = v4, T(v4) = v1

The j'th column of the matrix will consist of the coefficients of T(v_j) when expressed as a linear combination of the four basis vectors.

For example, $Tv_1=v_2=0v_1+1v_2+0v_3+0v_4$ , so the first column of the matrix will consist of 0,1,0,0.

Thus the matrix will be $\begin{bmatrix}0&0&0&1\\1&0&0&0\\0&1&0&0\\0&0&1&0\ end{bmatrix}.$

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