Which of the subsets V of given in Exercise 6 through 11 are subspaces of . The diagonal 3x3 matrices.
The diagonal 3x3 matrices subset V of is a subspace of .
A subset W of a linear space V is called a subspace of V if
(a) W contains the neutral element 0 of V.
(b) W is closed under addition (if f and g are in W then so is f+g )
(c) W is closed under scalar multiplication (if f is in W and k is scalar, then kf is in W).
we can summarize parts b and c by saying that W is closed under linear combinations.
Consider two 3x3 matrix namely,
Thus, the sum of two diagonal matrices is a diagonal matrix, because all other entries will be zero and only the entries in the diagonal can be added and remain non-zero elements.
Similarly, when multiplying a matrix with a scalar, only the non-zero elements in the diagonals can be multiplied and all other will remain as zeros.
Hence, diagonal matrices are a subspace of .
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