Question: 4. Let on > 0 be the maximum singular value of A e Rmxx. Suppose v E R is the unit 2-norm vector such that

4. Let on > 0 be the maximum singular value of A e Rmxx. Suppose v E R" is the unit 2-norm vector such that max 1| Ac | |2. TER", |:2=1 Show that 01 = || Av1ll2 = 1/All2. 5. For a matrix A e Rmxa, let of > 0 be its maximum singular value. Suppose ", E Rm, v1 ( R" be vectors such that |212 = 1012=1, All2 = 01, Av = 0141. If U E Rmxm and V E Roxn are orthogonal matrices such that the first column of U is uj and the first column of V is v1, show that UT AV = 01 0 0 B for some matrix B ER(m-1)*(-1). In the above, each 0 denotes a row or column zero vector

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( 1 + x ) ( 1 - x + x ^ 2 - x ^ 3 + x ^ 4 ) = 1 + a ^ 5