In each of problem, determine whether the vectors are linearly independent or dependent in the appropriate R. 10.4.10. <1,0,0,0 >, < 0, 1, 1,0

In each of problem, determine whether the vectors are linearly independent or dependent in the appropriate R. 10.4.10. , < 0, 1, 1,0 >, < -4,6,6,0> in R 10.4.13. , < 4,1 >, < 6,6> in R In each of problem, show that the set S is a subspace of the appropriate R" and find a basis for this subspace and its dimension. 10.4.17. S consists of all vectors < x, y, -y, -x > in R 10.4.20. S consists of all vectors in R6 of the form < x, x, y, y, 0, z> 10.4.23 Verify that the given vectors form a basis for the subspace S of R" that they span. Show that the given vector X is in S by writing it as a linear combination of these basis vectors. X = < -5, -3, -3>, basis vectors < 1, 1,1 >. < 0,1,1> 10.4.29 Suppose we are given a finite set of vectors in R", and one vector is the zero vector. Show that this set of vectors must be linearly dependent. 10.5.1 Let V1, V be mutually orthogonal vectors in R". Prove that ||V + ... + Vk || = ||V|| + ... + ||V || Hint: Write ||V + ... + V|| = (V + ... + Vk) (V + ... + Vk) 10.5.3 Suppose V1, ..., Vn form an orthonormal basis for R". Let X be any vector in R". Show that 72 (X-V) = ||X|| j=1 This is called Parseval's equality for vectors. 10.6.1 Written u as the sum of a vector in S and a vector in St. Also find the distance between u and S. u = < -2,6, 1,7 > and S has orthogonal basis: < 1,-1,0,0 >, < 1,1,0,0 > 10.6.8 Let S be the subspace of R spanned by < 1,0, 1,0 > and < -2,0, 2,1 >. Find the vector in S closest to < 1,-1, 3, -3 >.