Download 15 Questions on Linear Algebra - Sample Paper | MATH 4435 and more Papers Linear Algebra in PDF only on Docsity! Math 4435/6635 Linear Algebra 1 Georgia State University (This paper consists of pages.) Test II due October 15, 2003 Points: A ≥ 91+ Last name: POINTS First name: Calculators are not needed or permitted. Write neatly. Place answers in the space provided. The full credit is given only if all intermediate calculations and the entire work are shown. ¨¨¨¨ Problems for 6000 level are marked with asterisks 1 (15 points). ∗ ∗ ∗ ∗ ∗∗ (a) Let v 6= 0 in Rn. Show that vTy = vTx does not necessary imply that y = x. Draw the figure. (b) Show that if uTx = uTy for all unit vectors then y = x. Math 4435/6635 Linear Algebra 2 2 (10 points). ∗ ∗ ∗ ∗ ∗ ∗ ∗ Show that u ⊥ v in Rn iff ||v + u|| = ||v− u|| Math 4435/6635 Linear Algebra 5 6 (5 points). Let W = Span{v1, . . . ,vn}. Show that if x is orthogonal to each vi, 1 ≤ i ≤ n, then x is orthogonal to every vector in W . 7 (10 points). ∗ ∗ ∗ ∗ ∗∗ Given u 6= 0 in Rn, let L = Span{u}. Show that the mapping v → 2u Tv uTu v− v is a linear transformation. Draw the figure. Define this transformation. Math 4435/6635 Linear Algebra 6 8 (10 points). Let B be a square matrix with orthonormal (i.g. orthogonal and normalized) columns. Show that B is invertible. Mention all theorems you use. 9 (10 points). Show that the mapping x → Bx with such matrix B above preserves the norm of the vector x. Math 4435/6635 Linear Algebra 7 10 (5 points). Find the point on the line x + 2y − 1 = 0 closest to the point P (2, 1). Math 4435/6635 Linear Algebra 10 13 (10 points). ∗∗∗∗∗∗ Let A be an m × n matrix with linearly independent columns. Use the normal equation AT Ax̂ = ATb to produce a formula for b̂ — a vector projection of b onto R(A). Hint: find x̂ first. Math 4435/6635 Linear Algebra 11 14 (10 points). ∗∗∗∗∗∗∗ Verify that the error vector b−Ax̂, where x̂, is orthogonal to the column space of the matrix A; here A = 1 1 −1 1 −1 2 b = 7 0 −7 Math 4435/6635 Linear Algebra 12 15 (15 points). ∗ ∗ ∗ ∗ ∗∗ Find the equation y = c1x + c2x2 of the least square quadratic curve that best fits the given data: x -1 0 1 2 y 1 0 2 5 Put the points and draw the found curve in (x, y)-plane. FFF Must!!! Double-check yourself — solve this problem using the calculus approach based on minimization of a function of two variables.
