FIND THE DIMENSIONS OF THE FOLLOWING VECTOR SPACES.

a) The dimension is 4+3+2+1=10. There are 10 entries of upper triangular 4 by 4 matrices. Thus the measurement is 10.

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b) The measurement is 5. There space 5 entries of diagonal 5 by 5 matrices. Because of this the dimension is 5.

c) The dimension is 6x7=42.

d) The measurement is 3.

e) The measurement is 3. Polynomial of degree no much more than 2 is of the type ax2+bx+c, wherein a,b,c are any type of

real numbers. Therefore the measurement of the room is 3.

f) The dimension is 5x5-1=24. I think you have actually a typo here. It need to be traceless. Otherwise the space is not a vector space. The dimension of an are of 5 by 5 matrices is 25. The space of traceless matrices is a subspace with one measurement less, because any kind of traceless procession satisfies the condition that the amount of diagonal entries is 0.

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