Homework # due Thursday, Oct. 0. Show that the diagonals of a square are orthogonal to one another. Hint: Place the vertices of the square along the axes and then introduce coordinates. 2. Find the equation of the plane which contains A(,, 4), B(2, 2, ) and C(4, 0, 2).. Find the equation of the plane which contains both the point (, 2, ) and the line x = 2 t, y = + t, z = 5 + 4t. 4. Consider the line through (, 2, ) which is perpendicular to the plane 2x + y + 4z = 6. At which point does this line intersect the plane x 2y + z = 0?
Homework #2 due Thursday, Oct. 7. Find the distance between the point A(, 2, 4) and the plane 2x + y + 2z = 6. 2. Find a quadratic polynomial, say f(x) = ax 2 + bx + c, such that f() = 6, f(2) =, f() = 26.. Solve the system of linear equations 2x 2y + 2z = 6 x 4y + 2z = 4. 2x + y + 2z = 4. Solve the system of linear equations x + 2x 2 + 4x + 5x 4 + 6x 5 = 2 2x + x 2 + 5x + 7x 4 + 9x 5 = 7. 2x + 2x 2 + 6x + 8x 4 + 9x 5 = x + 5x 2 + 7x + 2x 4 + x 5 = 5
Homework # due Thursday, Oct. 24. Express w as a linear combination of u, u 2 and u in the case that 4 2 u = 0, u 2 = 0 2, u = 2, w = 5. 2 0 2. Show that a system of m linear equations in n > m unknowns cannot have a unique solution. Hint: count the pivots and the rows of the reduced row echelon form.. The trace of an n n matrix A is the sum of its diagonal entries, namely tr A = A + A 22 +... + A nn = n A kk. k= Show that tr(ab) = tr(ba) for all n n matrices A, B. 4. Suppose A, B are n n matrices and A has a row of zeros. Show that AB has a row of zeros as well and conclude that A is not invertible.
Homework #4 due Thursday, Oct.. Compute the determinant of the matrix 2 A = a 2. 2 a 2. Find the inverse of the matrix 2 A = 2 2. 2. Suppose A is a matrix whose third row is the sum of the first two rows. Show that A is not invertible and find a vector b such that Ax = b has no solutions. Hint: use row reduction for the first part; write down the equations for the second part. 4. Let A n denote the n n matrix whose diagonal entries are equal to and all other entries are equal to. Show that A n is invertible for each n. Hint: if you add the last n rows to the first row, then row reduction becomes somewhat easier; work out the cases n = 2, first.
Homework #5 due Thursday, Nov. 4. Compute det A using (a) expansion by minors and (b) row reduction: a A = 2. 2 a 2 2. Compute the adjoint and the inverse of the matrix A = 2. 2 2 5. Suppose A is an invertible n n matrix. Express det(adj A) in terms of det A. 4. Suppose A is a lower triangular matrix whose diagonal entries are all nonzero. Show that A is invertible and that its inverse is lower triangular. Hint: Tutorial problems #2 should be useful for the first part; the second part is related to the adjoint of A.
Homework #6 due Thursday, Nov. 2. Suppose that P is an n n permutation matrix. Show that P P t = I n. 2. The determinant of a 9 9 matrix A contains the terms a 8 a 29 a 7 a 4 a 52 a 6 a 76 a 84 a 95, a a 28 a 6 a 49 a 52 a 6 a 77 a 85 a 94. What is the coefficient of each of these terms?. Determine both the null space and the column space of the matrix 4 6 6 A = 2 4 4. 2 5 4 4. Suppose that A is a square matrix whose column space is equal to its null space. Show that A 2 must be the zero matrix.
Homework #7 due Thursday, Nov. 28. Suppose that the vectors v, v 2,..., v k form a complete set in R n and that they are linearly independent. Show that k = n and that the matrix whose columns are these vectors is invertible. 2. Is the matrix A a linear combination of the other three matrices? Explain. 4 9 2 A =, B 8 5 =, B 2 =, B 2 =. 2. Show that the following matrices are linearly independent in M 22. 0 0 0 A =, A 0 2 =, A =, A 0 4 =. 0 4. Suppose u, v, w are linearly independent vectors of a vector space V. Show that the vectors u, u + v, u + v + w are linearly independent as well.
Homework #8 due Thursday, Dec. 5. Let U be the set of all polynomials f P such that f(0) = f(). Show that U is a subspace of P and find a basis for it. 2. Show that v, v 2, v form a basis of R and then find the coordinate vector of v with respect to this basis when v = 2, v 2 =, 2 v =, 7 v = 5. 5. Show that w, w 2 form a basis of R 2 when 2 w =, w 2 =. Compute the coordinate vectors of e and e 2 with respect to this basis. 4. Let w, w 2 be as above. Find a linear transformation T : R 2 R 2 such that T (w ) =, T (w 5 2 ) =. 9 Hint: express each of e, e 2 as a linear combination of w, w 2 and then use linearity to determine each of T (e ), T (e 2 ).