All vectors are in rn. check the true statements below:

(a) the orthogonal projection of y onto v is the same as the orthogonal projection of y onto cv whenever c≠0.
(b) if the columns of an m×n matrix a are orthonormal, then the linear mapping x→ax preserves lengths.
(c) if a set s={} has the property that ui⋅uj=0 whenever i≠j, then s is an orthonormal set.
(d) an orthogonal matrix is invertible.
(e) not every orthogonal set in rn is a linearly independent set.

A), B) and D) are true

Step-by-step explanation:

A) We can prove it as follows:

B) When you compute the product Ax, the i-th component is the matrix of the i-th column of A with x, denote this by Ai x. Then, we have that . Now, the colums of A are orthonormal so we have that (Ai x)^2=x_i^2. Then .

C) Consider . This set is orthogonal because , but S is not orthonormal because the norm of (0,2) is 2≠1.

D) Let A be an orthogonal matrix in . Then the columns of A form an orthonormal set. We have that . To see this, note than the component of the product is the dot product of the i-th row of and the jth row of . But the i-th row of is equal to the i-th column of . If i≠j, this product is equal to 0 (orthogonality) and if i=j this product is equal to 1 (the columns are unit vectors), then    

E) Consider S={e_1,0}. S is orthogonal but is not linearly independent, because 0∈S.

In fact, every orthogonal set in R^n without zero vectors is linearly independent. Take a orthogonal set and suppose that there are coefficients a_i such that . For any i, take the dot product with u_i in both sides of the equation. All product are zero except u_i·u_i=||u_i||. Then then .  

squares, rhombus, rectangles