Find the dimension of the following real vector spaces: (a) V = {A : A is m × n real matrices}. (b) V = {A : A is n × n real upper - triangular matrices}. (c) V = {A : A is n × n real symmetric matrices}
Question
Find the dimension of the following real vector spaces: (a) V = {A : A is m × n real matrices}. (b) V = {A : A is n × n real upper - triangular matrices}. (c) V = {A : A is n × n real symmetric matrices}
Solution
(a) The dimension of the vector space V, where A is an m × n real matrix, is m*n. This is because each entry in the m × n matrix can be considered as a dimension.
(b) The dimension of the vector space V, where A is an n × n real upper-triangular matrix, is n*(n+1)/2. This is because in an upper triangular matrix, the entries below the main diagonal are zero, so the dimensions are the n entries on the diagonal plus the n(n-1)/2 entries above the diagonal.
(c) The dimension of the vector space V, where A is an n × n real symmetric matrix, is also n*(n+1)/2. This is because in a symmetric matrix, the entries above the main diagonal are the same as those below it, so we only need to consider the n entries on the diagonal and the n(n-1)/2 entries above (or below) the diagonal.
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