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If the columns of a matrix are orthonormal, then the linear mapping preserves lengths.

B. Not every orthogonal set is a linearly independent set.

C. An orthogonal matrix is invertible.

D. If a set has the property that whenever it is non-empty, then it is an orthonormal set.

E. The orthogonal projection of a vector onto a subspace is the same as the orthogonal projection of the vector onto that subspace whenever the conditions are satisfied.

Answer :

Final answer:

Most of the given statements related to linear algebra are accurate- like the properties of orthonormal matrices, invertibility of orthogonal matrices, and orthogonal sets being linearly independent- but some lack context. The direction of projection matters in orthogonal projections.

Explanation:

The question seems to be a mix of different statements generally related to the field of Mathematics -specifically dealing with linear algebra and concepts of vectors. Let's go through each statement:

  1. If the columns of a matrix are orthonormal, then the linear mapping preserves lengths. This is true, as orthonormal matrices have the property of preserving the dot product, and consequently, the lengths and angles between vectors.
  2. Not every orthogonal set in is a linearly independent set. This statement is not correct. In fact, in linear algebra, an orthogonal set of non-zero vectors is always linearly independent.
  3. An orthogonal matrix is invertible. This is indeed true. An orthogonal matrix has inverses; actually, the transpose of an orthogonal matrix is its inverse.
  4. If a set has the property that whenever, then it is an orthonormal set. This statement needs clarification but generally, an orthonormal set of vectors is one in which all vectors are orthogonal (perpendicular) to each other and each vector has a length (norm) of one.
  5. The orthogonal projection of onto is the same as the orthogonal projection of onto whenever. Generally, the direction of projection matters for orthogonal projections but without additional context or specifics for this statement, it's hard to definitively say whether it's true or false.

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