Final answer:
The dimensions of (a) the space of all lower triangular 7×7 matrices is 28, (b) R ⁶x⁴ is 24, and (c) the space of all diagonal 3×3 matrices is 3.
Step-by-step explanation:
(a) The space of all lower triangular 7×7 matrices: The lower triangular 7x7 matrices have the form:
[ a 0 0 0 0 0 0]
[ b c 0 0 0 0 0]
[ d e f 0 0 0 0]
[ g h i j 0 0 0]
[ k l m n o 0 0]
[ p q r s t u 0]
[ v w x y z a1 b]
Since each entry is independent, we can think of each entry as a separate vector space. The dimensions of these vector spaces are:
[ a]: 1
[ b, c]: 2
[ d, e, f]: 3
[ g, h, i, j]: 4
[ k, l, m, n, o]: 5
[ p, q, r, s, t, u]: 6
[ v, w, x, y, z, a1, b]: 7
Therefore, the dimensions of the space of all lower triangular 7x7 matrices are 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.
(b) R ⁶x⁴: R ⁶x⁴ represents the space of all 6x4 real matrices. Each entry in the matrix is a real number, so the dimension of each entry is 1. Therefore, the total dimension of this space is 6x4 = 24.
(c) The space of all diagonal 3×3 matrices: The diagonal 3x3 matrices have the form:
[ a 0 0]
[ 0 b 0]
[ 0 0 c]
Since each entry is independent, we can think of each entry as a separate vector space. The dimensions of these vector spaces are:
[ a]: 1
[ b]: 1
[ c]: 1
Therefore, the dimensions of the space of all diagonal 3x3 matrices are 1 + 1 + 1 = 3.