Find a basis of the subspace W:=A

[ak123456]

Homework Statement

find a basis of the subspace W:=A\in M2*2(R) : trace (A)=0 of the vector space M2*2 (R) and hence determine the dimension of W

Homework Equations

The Attempt at a Solution

trace(A) denote the the sum of the diagonal elements of the matrix A=aij
do i need to choose some vectors to form a basis ,hence to determine the dimension ?

 

[CompuChip]

Well, the elements of M_2x2(R) are 2x2 matrices. Recall that a basis of a vector space is a set of elements e₁, ..., e_n such that any element can be expressed as a linear combination of the e_i.

A general 2x2 matrix looks like
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
so a basis would be, for example,
e_{11} = \begin{pmatrix} 1 &amp; 0 \\ 0 &amp; 0 \end{pmatrix}, <br /> e_{12} = \begin{pmatrix} 0 &amp; 1 \\ 0 &amp; 0 \end{pmatrix}, <br /> e_{21} = \begin{pmatrix} 0 &amp; 0 \\ 1 &amp; 0 \end{pmatrix},<br /> e_{22} = \begin{pmatrix} 0 &amp; 0 \\ 0 &amp; 1 \end{pmatrix}<br />.

(Aside question for you: how do you express A as a linear combination of these four elements?)

 

[yyat]

I suggest you write the condition trace(A)=0 in terms of the four matrix elements. Then try to find the simplest examples of matrices which satisfy trace(A)=0 (matrices with only one or two nonzero elements), then try to construct a basis.