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Chapter 4: Q39E (page 177)

What is the dimensions of the space of all upper triangularn脳n matrices?

Short Answer

Expert verified

The solution is 鈭�k=1nk=nn+12.

Step by step solution

01

Explanation for the dimension of the space

If a linear space has a basis with n elements then all other bases of V, consists of n elements as well.

Also we say that n is the dimension of V

dim(V)=n

02

Solution for the dimensions of the space of all upper triangular  matrices

Consider generally an脳n matrix hasdata-custom-editor="chemistry" nn-12 off diagonal coefficients. Thus, the dimension of the sub algebra of the upper triangular matrices is equal to

data-custom-editor="chemistry" nn-12+n=n2-n+2n2=n2+n2=nn+12

Thus the summation of the dimension ofn脳n matrix becomes

鈭�k=1nk=nn+12

Hence, the solution鈭�k=1nk=nn+12

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