Math 410, Linear Algebra Part 1: Multiple choice, T F ( 120 Points) Which of the following sets of vectors are orthonormal for the Euclidean inner product? 1) (A) » (9) 2) 3) 2 (B) 02 2 (C) (69) 02 (D) 6 9 Which of the following vectors is the orthogonal projection of (1, 3, −2) on the subspace of R³ spanned by (1, 0, 3), (1, 1, 2)? (A) (8/11, 34/11, −10/11) (C) (-85, -35,-220) (B) (5/11, 35/11, -20/11) (D) (-8, -2, -22) Compute the angle between the following pair of vectors in M22 using the standard inner product on M, 10 = (69), B = (1 0) 22- A = (A) 0 = COS -1 60° (B) 0 = COS = = 90° 2 (C) = cos 45° (D) = cos A = 0° 4) 5) Given the equation of the line x = 8−t; y = 6+2t x = 8 − t, 6+2t, which of the following points is not on the line? (A) (6, 10) (B) (7,9) (C) (3, 16) (D) (8,6) What is the terminal point of a vector with initial point A (-1, 0) that is equivalent to u = (3,4)? (A) (3,3) (B) (4,4) (C) (3,4) (D) (2,4) 6) 7) Find the basis for the subspace of R³ 2x+3y-5z = 0 (A) span 2 -35 0 3 (B) span 3 3 H -2|| 5 0 3 3 -09--80 (C) span 2 5 3 (D) span 25 3 Find an orthonormal basis for the solution space of the homogeneous system of linear equations [x₁ − 2x2 + x3 = 0 (2x-2x2-x3 = 0 1 + √21 √14 √29 (A) span (B) span (C) span 3 21 14 29 3 2 21 span 21 (D) 8) What is the area of the triangle defined by the points (1, 3, 2), (2, 3, 1), and (2, 2, 3)? (A) 12√√6 (B) √6 (C) 0 (D) 12 9) Find a, b, and c if (4,b,c) = a(2,3,2) (A) a = 4,b=6, c = 4 (B) a = 4,b=3, c = 2 (C) a = 2,b=6,c = 4 (D) a = 4,b=3, c = 4 10) If A is a 3×3 matrix and |4|=7, what is |24| (A) 72 (B) 14 (C) 42 (D) 56 11) Find all values of λ for which 12) 13) 14) (1–λ 1 0 0 2-2 0 0 −1 4-2 |= 0 If | 4| = −4, find all possible values of k, where (A) 3, -4 1 k A = k 3k (B) 0, -3 (C) -4, -1 (D) 4,-1 Which of the following statements best describes the following augmented matrix? 1 2 6 5 −1 1 -23 4 _ 21 (A) A is consistent with a unique solution. (B) A is consistent with infinitely many solutions. (C) A is inconsistent. (D) none of the above. If the matrix A is 4 × 2, B is 3 × 4, C is 2 × 4, D is 4 × 3, and E is 2 × 5, which of the following expressions is not defined? (A) (C) 15) 16) ATD + CBT CA+CBT (B) (B+DT)A (D) BDAE Which of the following matrices is not an elementary matrix? 1 00 1 0 0 1 23 0 A = 0 1 0 B 7 1 0C=0 1 0 D = i) -15 0 0 -1) 0 1 (B) C (C) B (D) A (A) D Which matrix will be used as the inverted coefficient matrix when solving the following system? 3x₁ + x2 = 4 5x₁ + 2x2 = 7 2 (A) (33) 3 -5 (C) - 1 2 17) 2 1 (B) 5 -3 2 (D) 5 (3) What value of b makes the following system consistent? 4x₁ + 2x2 = b 2x1 + x2 = 0 (A) b=-1 (B) b=0 (C) b=1 (D) b=2 18) If A is an invertible 3×3 matrix and |24ª¹| = 9, what is (金) (A) 8/27 (B) 1/9 (C) 8/9 (D) 9/2 19) Use matrix multiplication to find the image of the vector (2, 1) when it is rotated counterclockwise about the origin through an angle 0 = 45°. 3√2 (A) NSNS 3√√2 2 (B) (C) √2 √3 (D) 3√√√2 2 2 ~/N/S Calculate the distance between the points (3, 1, −2) and the plane 2x + y − 2z = 1 20) (A) 2/9 21) (B) 3/5 (C)10/3 (D) 2/3 Find an equation for the plane containing the point (1, 1, 3) that is perpendicular to the line x = = 2 − 3t; y = 1 + t; z = 2t (A) 3x-y-2z+4=0 (B) 3x-y-2z+4=0 (C) -3x+y-2z = 7 (D) 2x+y+3z-4=0 22) Which system is the associated normal equation to the following system? X1 1 2 = X2 5 3 8 X1 18 (A) = 7 X2 6 27 17 x1 29 (B) = 17 13 X2 20 9 X1 (C) = X2 23) (D) x1 20 = 20 Which of the following vectors is the orthogonal projection of (1,3,-2) on the subspace of R³ spanned by {(1,0,3), (1,1,2)}? (A)(음,,一) 35 (B) (11, 1, −20) (C) (-85, -35, -220) (D) (-8, -2, -22) Which of the following matrices is negative definite? 24) 3 0 0 0 3 -6 (A) 0 0 0 (B) 1 0 (C) (D) 1 0 -17 0 0 25) Which of the following matrices is orthogonal? Го 0 2 0 (A) 0 (B) 0 (C) 1 0 −1 1 1 St. (D) 0 320 0 45 3545