# Diagonal Matrix Representation, Standard Basis Vectors, Kernels and Linear Mapping

A linear mapping T : R2 -> R3 is defined on the standard basis vectors via

T(e1) = (1, 0, 1) and T(e2) = (1, 0, -1)

1. Calculate T(3,3)

2. Find the dimension of the range of T and the dimension of the kernel of T.

3. Find the matrix representation of T relative to the standard bases in R2, R3.

4. Find bases {v1, v2} for R2 and {w1, w2, w3} for R3 with respect to which T has diagonal matrix representation.

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A linear mapping T : R2 -> R3 is defined on the standard basis vectors via

T(e1) = (1, 0, 1) and T(e2) = (1, 0, -1)

1. calculate T(3,3)

Since T is linear, then T(3e1+3e2)=3T(e1)+3T(e2), therefore

T(3,3)＝T(3e1+3e2)=3T(e1)+3T(e2)= 3(1, 0, ...

#### Solution Summary

Diagonal Matrix Representation, Standard Basis Vectors, Kernels and Linear Mapping are investigated.