PLEASE COMPLETE ALL OF THE QUESTIONS!!! Determine whether the statements are tru

Question

PLEASE COMPLETE ALL OF THE QUESTIONS!!!

Determine whether the statements are true or false. a) The rows of an n times n orthogonal matrix form an orthonormal basis for R". b) If T: R^n rightarrow R^n is a function such that ||T (u) - T (v)|| = ||u - v|| for all vectors u and v in 'R^n, then T is an orthogonal operator. c) Every linear operator preserves dot products. d) If a linear operator preserves dot products, then it preserves norms. e) If P is an orthogonal matrix, then P^T is an orthogonal matrix. f) If P and Q are n times n orthogonal matrices, then PQ^T is an orthogonal matrix. g) If P and Q are n times n orthogonal matrices, then P + Q is an orthogonal matrix. h) If P is an n times n matrix such that del P = plusminus 1, then P is an orthogonal matrix. i) If P and Q are n times n orthogonal matrices, then PQ is an orthogonal matrix. j) If the columns of an n times n matrix form an orthogonal basis for R^n, then the matrix is an orthogonal matrix. k) For any subspace W of R^n, the matrix P_w is an orthogonal matrix. I) If P is a matrix such that P^T = P^-l, then P is an orthogonal matrix. m) Every orthogonal matrix is invertible.

Explanation / Answer

(a) true

(b) false

(c) false

(d) true

(e) true

(f) true

(g) true

(h) false

(i) true

(j) false

(k) false

(l) true

(m) true

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