6. (5%) Apply "Cauchy-Schwarz Inequality" and prove \| u +v\| ≤ \| u\| +\| v\|

(a) (5%) If \( L \) is a linear transformation mapping from \( R^n \) into \( R^m \), there is an \( m \) by \( n \) matrix \( A \) such that \( L(\vec{x}) = A\vec{x} \) for each \( \vec{x} \in R^n \). This is the matrix representation for a linear transformation. Prove that the \( j \)th column vector \( \vec{a}j \) of \( A \) is given by \( \vec{a}j = L(\vec{e}_j) \), for \( j = 1, 2, \dots, n \).

6. (5%) Apply "Cauchy-Schwarz Inequality" and prove \( \| \vec{u} +\vec{v}\| \leq \| \vec{u}\| +\| \vec{v}\| \)

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