Find the dimensions of the following vector spaces

Question

Find the dimensions of the following vector spaces.

(a) The vector space of all upper triangular 4x4  matrices 

(b) The vector space of all diagonal 5x5  matrices 

(c) The vector space  R^(6x7)

(d) The vector space R^3

 (e) The vector space P2(x) of polynomials with degree less than 2

 (f) The vector space of 5x5  matrices with trace 

Answer 1

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

b) The dimension is 5. There are 5 entries of diagonal 5 by 5 matrices. Therefore the dimension is 5.

c) The dimension is 6x7=42.

d) The dimension is 3.

e) The dimension is 3. Polynomial of degree no more than 2 is of the form ax²+bx+c, where a,b,c are any

real numbers. So the dimension of the space is 3.

f) The dimension is 5x5-1=24. I think you have a typo here. It should be traceless. Otherwise the space is not a vector space. The dimension of space of 5 by 5 matrices is 25. The space of traceless matrices is a subspace with one dimension less, because any traceless matrix satisfies the condition that the sum of diagonal entries is 0.