qsn 4 c and h Determine if the following is a subspace of the vector space R. A_

Question

qsn 4 c and h

Determine if the following is a subspace of the vector space R. A_m times n X_n times 1 - b_m times 1 A_m times n X_n times 1 = 0 Determine if the following is a subspace of the vector space M_3 times 3: a. U = { m | m is upper triangular } b. U = { m | all diagonal elements of m are equal to 1 } c. U = { m | m is invertible } d. U = { m | m is symmetric } e. U = { m | m has only integer components } f. U = { m | tr(m) = 0 } g. U = { m | m is nonsingular } h. U = { m | m is a diagonal matrix } i. U = { m | m is an orthogonal matrix }

Explanation / Answer

c)

The set of all invertible n×nn×n matrices of real numbers is NOT a subspace

Let for example I, the unit matrix is invertible and so is I. But their sum I+(I)=0 is definitely not invertible!

h) Yes

as v1 = [d1 ; 0 0 ; 0 d2 0 ; 0 0 d3] ;

v2 = [a1 0 0 ; 0 a2 0 ; 0 0 a3];

v1+ v2 = [ a1+ d1 ; 0 0 ; 0 b2+ d2 0 ; 0 0 b3+ d3] ;

hence v1+ v2 is also diagonal matrix

cv = [c*d1 ; 0 0 ; 0 c*d2 0 ; 0 0 c*d3] ;

cv is also diagonal matrix

hence

it is subspace

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