Consider a subspace in that is defined by homogeneous linear equations
What is the relationship between the dimension of and the quantity
? State your answer as an inequality. Explain carefully.
A subspace of has dimension at least .
We have the subspace of defined by homogeneous linear eqautions,
role="math" localid="1660126682574" ,
Therefore, can be written as
, where is an matrix .
Suppose the column vector be .
Theorem 3.3.7, is stated as follows:
For any matrix , the equation,
We have the rank of .
Thus, by Theorem 3.3.7, we have,
Hence, a subspace of has dimension at least .
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