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a)

Consider a counterexample that \({\bf{y}} = \left( {\begin{array}{*{20}{c}}7\\6\end{array}} \right)\)and \({\bf{u}} = \left( {\begin{array}{*{20}{c}}4\\2\end{array}} \right)\). Then, \(\left\{ {{\bf{y}},{\bf{u}}} \right\}\) is linearly independent sets but not an orthogonal set because \({\bf{y}} \cdot {\bf{u}} \ne 0\).

Thus, the given statement (a) is true.

b)

Theorem 5states that consider \(\left\{ {{{\mathop{\rm u}\nolimits} _1},{{\mathop{\rm u}\nolimits} _2}, \ldots ,{{\mathop{\rm u}\nolimits} _p}} \right\}\) as anorthogonal basisfor a subspace \(W\) of \({\mathbb{R}^n}\), then theweightsin the linear combination \({\mathop{\rm y}\nolimits} = {c_1}{{\bf{u}}_1} + \cdots + {c_1}{{\bf{u}}_p}\)for every y in \(W\) is denoted by \({c_j} = \frac{{{\bf{y}} \cdot {{\bf{u}}_j}}}{{{{\bf{u}}_j} \cdot {{\bf{u}}_j}}}\left( {j = 1, \ldots ,p} \right)\).

Thus, the given statement (b) is true.

c)

If the nonzero vector in an orthogonal set isnormalized to have unit length, then the new vectors remain orthogonal.

Thus, the given statement (c) is false.

d)

Asquare invertible matrix\(U\)is an orthogonal matrix, such that \({U^{ - 1}} = {U^T}\). Such a square matrix contains orthonormal columns.

The square matrix with orthonormal columns is orthogonal.

Thus, the given statement (d) is false.

e)

It is known that the distance fromy to L is equal to the length of the perpendicular line segment from yto the orthogonal projection \(\widehat {\bf{y}}\), that is \(\left\| {{\bf{y}} - \widehat {\bf{y}}} \right\|\).

Thus, the given statement (e) is false.