QuestionNYC 05 VectorSpaces: Problem 12
(1 point)
(a) If is the subspace of consisting of all upper triangular matrices, then
(b) If is the subspace of consisting of all matrices with trace 0 , then
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Studdy Solution
STEP 1
1. denotes the space of all matrices with real entries.
2. The dimension of a subspace is the number of vectors in a basis for that subspace.
3. An upper triangular matrix has all zeros below the main diagonal.
4. The trace of a matrix is the sum of its diagonal elements.
STEP 2
1. Determine the dimension of the subspace of upper triangular matrices in .
2. Determine the dimension of the subspace of matrices with trace 0 in .
STEP 3
To find the dimension of the subspace of upper triangular matrices in , count the number of free entries in an upper triangular matrix.
An upper triangular matrix in has the form:
There are free entries (the diagonal and above).
STEP 4
To find the dimension of the subspace of matrices with trace 0 in , consider the condition that the sum of the diagonal elements is zero.
A general matrix has 16 entries. The trace condition imposes one linear constraint (i.e., ), reducing the number of free entries by 1.
Thus, the dimension is .
The dimensions are:
(a)
(b)
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