What is the maximum number of orthonormal vectors we can find in R^n? b) True or

Question

What is the maximum number of orthonormal vectors we can find in R^n? b) True or false: if dim(V) = n, and v_1, ..., v_n is a collection of orthogonal vectors in V (and none of the vectors are zero), then V = . c) True of false: If Q is a p times n matrix with orthonormal columns, then p greaterthanorequalto n. d) True or false: if Q is a square matrix and it has orthonormal columns, then it has orthonormal rows. e) True or false: if Q is a rectangular matrix (meaning not a square matrix) with orthonormal columns, then it has orthonormal rows. f) True or false: if Q is an orthogonal matrix, then Q^-1 is an orthogonal matrix. g) True or false: if Q is an orthogonal matrix, then for any vector v, ||Q_v|| = ||v||.

Explanation / Answer

The maximum number of orthonormal vectos in R2 is 2 and in R3 , this number is 3. In Rn, the maximum number of orthonormal vectors is n. The statement is true. The orthogonal vectors are linearly independent and hence V = Span{v1,v2,…,vn}. The statement is true. We can not have a 2x3 matrix with orthonormal columns, but a 2x2 or 3x2 or 3x3 matrix can have orthonormal coumns. The inference can be drawn by induction. The statement is true. If Q is a square matrix with orthonormal columns, then Q is an orthogonal matrix so that QT is also an orhogonal matrix. Hence the rows of Q(the columns of QT) are orthonormal. The statement is false(not necessarily true). For example, if Q has the vectors(1,0,0)T and(0,1,0)T as its columns, then the columns are othonormal, but the 3rd row is a zero vector. The statement is true. If Q is an orthogonal matrix , then Q-1 = QT is also an orthogonal matrix. The statement is true. An orthognal matrix preserves the lengths of vectors, i.e. = or, ||Qv||= ||v||.

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