QuestionNYC 05 VectorSpaces: Problem 12 (1 point) (a) If $S$ is the subspace of $M_{5}(\mathbb{R})$ consisting of all upper triangular matrices, then $\operatorname{dim} S=$ $\square$ (b) If $S$ is the subspace of $M_{4}(\mathbb{R})$ consisting of all matrices with trace 0 , then $\operatorname{dim} S=$ $\square$
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Studdy Solution
To find the dimension of the subspace $S$ of matrices with trace 0 in $M_4(\mathbb{R})$, consider the condition that the sum of the diagonal elements is zero.
A general $4 \times 4$ matrix has 16 entries. The trace condition imposes one linear constraint (i.e., $a_{11} + a_{22} + a_{33} + a_{44} = 0$), reducing the number of free entries by 1.
Thus, the dimension is $16 - 1 = 15$.
The dimensions are: (a) $\operatorname{dim} S = 15$ (b) $\operatorname{dim} S = 15$