24. Question: In Exercises 23 and 24, all vectors are in \({\mathbb{R}^n}\). Mark each statement True or False. Justify each answer.

1. Not every orthogonal set in \({\mathbb{R}^n}\) is linearly independent.
2. If a set \(S = \left\{ {{{\mathop{\rm u}\nolimits} _1}, \ldots ,{{\mathop{\rm u}\nolimits} _p}} \right\}\) has the property that \({{\mathop{\rm u}\nolimits} _i} \cdot {{\mathop{\rm u}\nolimits} _j} = 0\) whenever \(i \ne j\), then \(S\) is an orthonormal set.
3. If the columns of a \(m \times n\) matrix A are orthonormal, then the linear mapping \({\mathop{\rm x}\nolimits} \mapsto A{\mathop{\rm x}\nolimits} \) preserves lengths.
4. The orthogonal projection of y onto v is the same as the orthogonal projection of y onto \(c{\mathop{\rm v}\nolimits} \) whenever \(c \ne 0\).
5. An orthogonal matrix is invertible.

a)

Theorem 4states that when theorthogonal set of nonzerovectors in \({\mathbb{R}^n}\) is defined as \(S = \left\{ {{{\mathop{\rm u}\nolimits} _1},{{\mathop{\rm u}\nolimits} _2}, \ldots ,{{\mathop{\rm u}\nolimits} _p}} \right\}\), then the set \(S\) is linearly independent and, therefore, \(S\) is abasisfor the subspace spanned by \(S\).

However, the nonzero vectors of every orthogonal set are linearly independent according to theorem 4.

Thus, the given statement (a) is true.

b)

A set \(\left\{ {{{\mathop{\rm u}\nolimits} _1},{{\mathop{\rm u}\nolimits} _2}, \ldots ,{{\mathop{\rm u}\nolimits} _p}} \right\}\) is called anorthonormal set when it is an orthogonal set of unit vectors.

Thus, the given statement (b) is false.

c)

According toTheorem 7, consider that \(U\) as an \(m \times n\) matrix with orthonormal columns, and assume that \({\bf{x}}\) and \({\bf{y}}\) are in \({\mathbb{R}^n}\). Then,

1. \(\left\| {U{\bf{x}}} \right\| = \left\| {\bf{x}} \right\|\)
2. \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = {\bf{x}} \cdot {\bf{y}}\)
3. \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = 0\)such that if \({\bf{x}} \cdot {\bf{y}} = 0\).

Thus, the given statement (c) is true.

d)

When cbe any nonzero scalar and u in the definition of \(\widehat {\bf{y}}\) is replaced by \(c{\bf{u}}\), then the orthogonal projection of y onto \(c{\bf{u}}\) is precisely the same as the orthogonal projection of y onto u.

Thus, the given statement (d) is true.

e)

Asquare invertible matrix\(U\)is known as an orthogonal matrix as such \({U^{ - 1}} = {U^T}\). This matrix contains orthonormal columns.

Thus, the given statement (e) is true.