What is the dimension of the subspace $$ W=\left\{A=\left[a_{i j}\right] \in \mathbb{R}_{4 \times 2} \mid a_{42}=0\right\} $$ ? $$ \operatorname{dim} W=\text { Ex:5 } $$

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Let $$ A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \ a_{31} & a_{32} \ a_{41} & a_{42} \end{pmatrix} $$ be an arbitrary matrix in $$\mathbb{R}^{4 \times 2}$$.

1. The total number of entries in $$ A $$ is $$4 \times 2 = 8$$.

2. The subspace $$ W $$ is defined by the condition $$ a_{42} = 0 $$. This condition fixes one entry of the matrix, leaving the remaining entries free.

3. Therefore, the number of free entries (degrees of freedom) is $$8 - 1 = 7$$.

Thus, the dimension of the subspace $$ W $$ is

$$
\operatorname{dim} W = 7.
$$
The dimension of the subspace $$ W $$ is 7.

What is the dimension of the subspace ?

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