Find a basis of the subspace W:=A

  • Thread starter ak123456
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Homework Statement


find a basis of the subspace W:=A[tex]\in[/tex] 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 ?
 

Answers and Replies

  • #2
CompuChip
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Well, the elements of M2x2(R) are 2x2 matrices. Recall that a basis of a vector space is a set of elements e1, ..., en such that any element can be expressed as a linear combination of the ei.

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

(Aside question for you: how do you express A as a linear combination of these four elements?)
 
  • #3
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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.
 

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