Show transcribed image text 1. [10 pointsl Let A=12 5 1-1 i) Find the columns of A which constitute a basis of its column space;ii Write the nonbasis columns of A into linear combinations of the basis columns; i) Find the nullspace of A and determine a basis. iv) Find the dimensions of the row space and column space. 3. [10 points] i) Show that N(A) = N(EA) if E is invertible. ii) Construct an example of a 2 x 2 matrix A and an elementary matrix E such that C(A-C(EA). 4. [10 points] Let A be i) Find all possible vectors x such that x E R(A)nN(A), where R(A) denotes the row space of A. ii) Let m> n. Show that AAT is not invertible. 5. [10 points] Let A = | 4 5 6|. i) Find N(A) and split x = (1, 1, 1) into x = zr+Zn with In EN(A) and xrE R(A); Find a basis of the orthogonal complement of N(A) 、 [10 points] Let β Rn be a nonzero vector. Show that ī) V = {xER'. : x . β = 0} is a subspace of R": ii) dim V = n-1. 7. [10 points] Let A E Mnn be such that A2A and ATA. Show that i) N(A) = C(I-A); ii) C(A) C(1-A), where C(A) denotes the column space of A; iii) every x Rn can be uniquely decomposed as x = xc+x, where xeE C(A) and Xi C(1-A).