= 9. Let {x1, X2, ..., Xn-1} be a set of linearly independent vectors in R", and let W Span{X1, X2,...,xn-1}. Prove that if x ER", then x EW if and only if det (x xi ... Xn-1] = 0.

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8. Determine if each of the following statements is true or false. Clearly indicate your answer for each part by stating if it is true or false no justification is required). Each part is worth 2 marks for a correct answer; -1 for an incorrect answer to discourage guessing) (a) If A and B are n x n matrices, then rank(BA) rank(A). (b) If {A, B, C} is a linearly independent set of n x m matrices, then {AT, BT,CT} is a linearly independent set of m x n matrices. (C) W = {A E Rnxn A is invertible } is a subspace of Rnxn, where Rnxn denotes the vector space of real n x n matrices. (d) If xi and X2 are orthogonal vectors in R3 then it is always possible to find a unique vector X3 such that {x1, X2, X3} is an orthogonal basis for R3. (e) Let W be a subspace of R" and let x ER". Then projw(x - projw(x)) = 0.

6. Let x, y, z) be a linearly independent set of vectors in R". Prove or disprove the following statement: The set x-y,y - Z, Z-x} is also linearly independent. [5 marks)

4. Let S= 1 1 , and consider the subspace W = Span(S) of R. (a) Show that S is an orthogonal basis for W. (b) Let x = 2. Find projw(x). 11 (c) Let x = 12 be an arbitrary vector in R3. Find projw(x) and then find a matrix A such that projw(x) = Ax.

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= 9. Let {x1, X2, ..., Xn-1} be a set of linearly independent vectors in R", and let W Span{X1, X2,...,xn-1}. Prove that if x ER", then x EW if and only if det (x xi ... Xn-1] = 0.

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