(Solved): 1. A \( 5 \times 1 \) vector \( v \) of length 1 is transformed by \( A \) as \( u=A v \). What is ...

1. A \( 5 \times 1 \) vector \( v \) of length 1 is transformed by \( A \) as \( u=A v \). What is the longest length that \( u \) can be? 2. What is the shortest non-zero length of \( u \) ? 3. What is the rank of \( A \) ? The null space of \( A \) is the space of all vectors \( v \) such that \( A v=0 \). 4. What is the dimensionality of the null space of \( A \) ? I) Answer the above four questions for the following matrix. Here \( B \) transform 4-D vectors to 5-D. Note that \( B=A^{T} \). \[ B=\left[\begin{array}{cccc} 1 & 2 & 2 & 4 \\ 1 & 3 & 1 & 7 \\ 2 & 4 & 5 & 9 \\ 3 & 5 & 6 & 8 \\ 4 & 7 & 11 & 15 \end{array}\right] \]