Linear Algebra Answers

a) Find a matrix P that diagonalizes A and determine 𝑷

−𝟏𝑨𝑷, where

𝑨 = (

𝟐 −𝟏 𝟎 𝟎

−𝟐 𝟑 𝟎 𝟎

𝟐 𝟎 𝟒 𝟐

𝟏 𝟑 −𝟐 −𝟏

)

b) Let L denote the linear transformation in ℝ𝟐 which describes a reflection in 

ℝ𝟐

about the line 𝒙𝟐 = 𝒙𝟏. Find the matrix of A and its eigenvalues and 

eigenvectors.

c) The matrix of a linear transformation T on ℝ𝟑

relative to the usual basis 

{𝒆𝟏 = (𝟏, 𝟎,𝟎), 𝒆𝟐 = (𝟎,𝟏, 𝟎), 𝒆𝟑 = (𝟎,𝟎, 𝟏)} is [

𝟎 𝟏 𝟏

𝟏 𝟎 −𝟏

−𝟏 −𝟏 𝟎

]. Find the 

• matrix of T relative to the basis {(𝟎, 𝟏,𝟐), (𝟏, 𝟏,𝟏), (𝟏,𝟎, 𝟐)}.

 Let Mn(R) be the vector space of all n×n real matrices and W be the set of all +2)

n × n real matrices of zero trace. Show that W is a subspace of Mn(R). Find

a basis of W.

. Show that for any g 2 L(V; C) and u 2 V with g(u) 6= 0: V = null g  fu :  2 Cg.

Suppose V is finite-dimensional with dimV greater or equal 2. Prove that there exist S,T element of L(V,V) such that ST not equal TS.