MAE101
If U = (1; 1;-1); V = (0; 2;-1); W = (1;-3; 3); then the cosine of the angle between V x W and U x V is:
-1 / sqrt 21
Find a so that B = {(} is an orthonormal set.
1 / sqrt 6 or -1/ sqrt 6
Find the shortest distance between the pair of nonparallel lines [x y z]T = [1 -1 -1]T+t[2 3 1]T and [x y z]T = [1 -2 -1]T+t[3 2 2]T .
1 / sqrt(42)
Find a so that B = {(} is an orthonormal set.
1/6 sqrt -1 /6
Let be projection on the line y = 2x followed by reflection in the x-axis and let the vector v = [1 1]T. Find T(v).
3/5 -6/5
The volume of the pyramid with vertices (0 ; 0 ; 0) ; ( - 1 ; 8 ; 1) ; ( -16 ; 0 ; 1) and (2 ; 0 ;-2) is:
Let be rotation through followed by reflection in the line . Find (2,1)-entry of the matrix of .
Consider the points A(1; 2; 3); B(1; 3; 2) and C(2; 1; 3). Find a point D on the Z-axis so that the volume of the tetrahedron ABCD is 3.
The correct answer is: (0, 0 -12)
Let u = (1, 2, 1); v = (2, 3, 0); w = (0, 1, 0). Find projection of u x v on w.
The correct answer is: (0, 2, 0)
Let T be projection on the line y = 2x followed by rotation through . Find T[x y].
The correct answer is: (1/5)[-2x-4y; x+2y]
Let T be projection on the line y = 2x followed by reflection in the X axis. Find T[x y].
The correct answer is: (1/5)[x+2y; -2x-4y]
Let P(1,2,1), Q(1,0,-1), R(2,2,0) be the vertices of a parallelogram with adjacent sides QP and QR. Find the other vertex S.
The correct answer is: (2, 4, 2)
Let the point P(2, -1, 0). Find the point Q on the plane x - y + z = 1 that is closest to P.
The correct answer is: (4/3, -1/3, -2/3)
Let X = [a, b, c]. Let U be the subspace spanned by the orthogonal basis{u = [1, 1, 1], v = [1, -1, 0], w = [1, 1, -2]}.Find the coefficient of u when expressing X as a linear combination of {u, v, w}.
The correct answer is: (a+b+c)/3
Which of the following are subspaces of R2 ?(i) {(x, y)| x+2y=0}(ii) {(x, y)| x+y2= 0}
The correct answer is: (i)
Which of the following are subspaces of R3?(i) {(x,y3,z5)| x, y, z are real numbers}(ii) {(x, y, z)| x+y3+z5=0}
The correct answer is: (i)
Which of the following is a subspace of R4?(i) {(a,b,c,d)| a=b=0}(ii) {(a,b,c,d)| a=1, b=0 and c+d=1}(iii) { (a,b,c,d)| a>0 and b<0}
The correct answer is: (i)
Let the set of vectors {u, v, w} in R3 be independent. Which of the followings are true? (i) {2u; 3v } is also independent. (ii) {u; v; w-u-v } is also independent.
The correct answer is: (i) and (ii)
Which of the following are subspaces of R2 (i) {(x,y): 2x+7y = 0} (ii) {(x,y): y=0}
The correct answer is: (i) and (ii)
Let A be a 3 x 5 matrix and let dim(null(A))=2. Which of the following statements are true? (i) All bases of the Col(A) have three vectors. (ii) dim(Row(A)) = 1
The correct answer is: (i) only
Let A is a 150x350 matrix. Which of the following statements are true?(i) dim(Null(A)) must be at least 200.(ii) dim(col(A)) + dim(row(A)) = 500
The correct answer is: (i) only
Which of the following are subspaces of R3?(i) {(x, y, z)| 2x-y+3z=0}(ii) (x, y, z)| xy=0}
The correct answer is: (i) only
Let the set of vectors {u, v, w} in R3 be independent. Which of the followings are true? (i) {u; v; u-v+w} is also independent. (ii) {u; v+w; u+v+w} is also independent.
The correct answer is: (i) ony
Let A be 4 x 7 matrix that has dim(Null(A)) = 5. Choose the correct statements:(i) A has exactly 5 independent columns(ii) A has exactly 2 independent rows
The correct answer is: (ii)
Let u = (u1, u2, u3); v = (v1, v2, v3); w = (w1, w2, w3). Which of the following statements are false? (i) . (ii) , where is an angle beetween and . (iii)
The correct answer is: (ii)
Let the set of vectors {u, v, w} in R3 be dependent. Which of the followings are true? (i) w must be a linear combination of u and v. (ii) There exists (a, b, c) with a2+b2+c2 not zero such that a.u + b.v + c.w = 0
The correct answer is: (ii) only
Which of the following are subspaces of R3 ? (i) {(x,y,z)| z = (x+y)2} (ii) {(x,y,z)| x=10z}
The correct answer is: (ii) only
Which of the following are subspaces of R3? (i) {(x, y, z)| xz > 0 or xz = 0} (ii){(x, y, z)| x=y}
The correct answer is: (ii) only
Which of the following statements are true? (i) if {X, Y} is orthogonal in Rn then {X, X+Y} is also orthogonal. (ii) if {X,Y} and {Z, W} are both orthogonal then {X,Y,Z,W} is also orthogonal. (iii) if {X, Y} is orthonormal then {X-Y, X+Y} is orthogonal.
The correct answer is: (iii)
Which of the following statements are true? (u, v in R3) (i) || -5u || = -5|| u || (ii) || u-v || = || u || -|| v || (iii) If u, v, u+v are nonzero and u and (u+v) are parallel, then u and v are also parallel.
The correct answer is: (iii)
Let u1 = [-2, 0, 1] , u2 = [3, 5, 6] , u3 = [-2, 6, -4] , x = [-2, -32, -19] If express the vector x as x=au1 +bu2+cu3 then find c.
The correct answer is: -2
Let be rotation through followed by reflection in the line . Find (2,1)-entry of the matrix of .
The correct answer is: -4/5
For what values of x are the vectors [1, -1, 2], [1, x, -4], [-1, 0, x] linearly dependent?
The correct answer is: -4; 1
Find the dimension of Null(A) for A [ 1 2 -9 / 2 8 -38 / 5 12 -65 ]
The correct answer is: 1
Find the dimension of the subspace U = {[x+2y+3z, -2x-4y-6z, 5x+10y+15z]| x, y, z are real numbers}.
The correct answer is: 1
Find the number a such that the set {[1 2 1 0]T, [1 -1 1 3]T, [2 -1 0 -1]T, [a b c 1]T } is orthogonal.
The correct answer is: 1
If we write X = [2 -3 2 7]T as a linear combination of the orthogonal basis of the subspace U = span{[2 -1 0 3]T ; [2 1 -2 -1]T} then the sum of coefficients equals
The correct answer is: 1
Let Q be the point on the plane x+y+z=1 that is closet to P(1,1,1). Find the first coordinate of Q.
The correct answer is: 1/3
Find the shortest distance between the pair of parallel lines [x y z]T = [1 1 1]T+t[3 0 4]T and [x y z]T = [0 1 0]T+t[3 0 4]T
The correct answer is: 1/5
Let be reflection in the x-axis followed by reflection in the line . Find the sum of all entries in the first column of the matrix of .
The correct answer is: 1/5
Find the dimension of the following subspace of R3U=span{[1, 3, -1, -3], [2, 4,1, 0], [1, 5, -4, -9]}
The correct answer is: 2
Find the dimension of the null space of the matrix
The correct answer is: 2
Find the dimension of the null space of the matrix A = [1 -2 -1 -1 / 0 1 4 1 / 1 -1 3 0 ]
The correct answer is: 2
Find the dimension of the subspace U = span{[1, 2], [2, -1], [0, 4], [1, -5]}.
The correct answer is: 2
Find the dimension of the subspace U = {[2x+y+z; 4x+2y+z; 6x+3y+z ]}.
The correct answer is: 2
Find the dimension of the subspace U = {[x, y, z, t] | x+4y-z = 0; x-2z+ t= 0}.
The correct answer is: 2
Find the dimension of the subspaceU={[a+3c, b, a+3c] | a, b, c in R}
The correct answer is: 2
Find the dimension of the subspaceU={[a+c, b+c, a+2c+b] | a, b, c in R}
The correct answer is: 2
Find the dimension of the subspaceU={[x, y, z, w]| x-2y+3z+4w = 0, 3x-5y+7z+8w = 0}
The correct answer is: 2
If x = au1 +bu2 then find a+b.u1 = [2, -4] , u2 = [12, 6], x =[-26, -38]
The correct answer is: 2
Let the point P(2, -1, 3). Find the third coodinate of the point Q on the plane x - 2y + z = 1 that is closest to P.
The correct answer is: 2
Let Q be the point on the plane x+y+z=1 that is closet to P(1,0,1). Find the first coordinate of Q.
The correct answer is: 2/3
Find the dimension of the null space of the matrix [1 -2 3 -3 -1 / -2 5 -5 4 -4 /-1 3 -2 1 -5 ]
The correct answer is: 3
Find the dimension of the subspace U = span{[1, 1, 1], [2, 5, 2], [1, 2, 3]}.
The correct answer is: 3
Find the dimension of the subspace spanned by the vectors{[1, 1, 1], [-1, 1, -1], [1, 1, 3], [0, 2, 1]}
The correct answer is: 3
For what value of k are the two planes 3kx+y-5kz+10=0, 2x-3y+z+12=0 orthogonal?
The correct answer is: 3
Let A be a 3 x 5 matrix. If dim(null(A))=2, then the dimension of the column space of A is
The correct answer is: 3
Let u = (3, 3, 6), v = (4, 4, 3), w = (-6, 3, 3) and x = (41, 5, 12). We can write x asx = au + bv + cw,where a, b, c are numbers. Find a.
The correct answer is: 3
Find all values of a so that the vector [5, 3, a] is in span{[3, 2, 0], [1, 0, 3]}
The correct answer is: 3/2
The volume of the pyramid with vertices (0 ; 0 ; 0) ; ( - 1 ; 8 ; 1) ; ( -16 ; 0 ; 1) and (2 ; 0 ;-2) is:
The correct answer is: 40
Find an equation of the plane which contains the point (2, 4, 3) and which is perpendicular to the planes x+2y-z=1, 3x-4y = 2
The correct answer is: 4x+3y+10z=50
The volume of the pyramid with vertices (0; 0; 0); (-2; 8; 14); (-6; 7;-3) and (4; 0; 2) is:
The correct answer is: 70
Which of the following points lie on the line x = 2-t; y = 1+t; z = 7-2t? (i) A(-3, 6, -3) (i) B(0, 3, 0)
The correct answer is: A only
Let A is a 100x200 matrix. Which of the following statements are true?(i) dim(Null(A)) must be at least 100.(ii) dim(null(A)) + dim(row(A)) = 200
The correct answer is: Both (i) and (ii)
Find the point of intersection (if any) of the following pair of lines: d1: x =3+t; y =-1+t; z = 2-t d2: x = 1+2s; y = 1; z = -2+3s
The correct answer is: It does not exist
Let Q be the point on the line x = 1+t, y = -2 + 3t, z = 1 - t that is closet to the point P(1,0,1). Find the first coordinate of Q.
The correct answer is: None of the other choices is correct
Which of the following are subspaces of R2 ? (i) {(x,y)| x=y2} (ii) {(x,y)| xy > 0 or xy = 0}
The correct answer is: None of the other choices is correct
Which of the following are subspaces of R3 ? (i) {(x,y,z)| z = 2x+3y+2} (ii) {(x,y,z)| x2+y2=z2}
The correct answer is: None of the other choices is correct
Which of the following sets are linearly independent?(i) {[1, 1, 1], [1, 0, 1], [-1, 1, -1]}(ii) {[1, 2], [3, 4], [-1, -1]}(iii) {[1, 2, 1], [1, 1, 1], [-1, 0, 0], [0, 0, 1]}
The correct answer is: None of the other choices is correct
If . Determine if T is projection on a line, reflection in a line, or rotation through an angle, and find the line or angle.
The correct answer is: None of the other choices is true
Which of the following points lie on the line with parametric equations x = 2+4t; y = 3-7t; z = 5t? P(10; -11; 10); Q(-2; 4; 5); R(-2; 10; -5)
The correct answer is: P and R
If : T: R2 -> R2 T[x y ] = [ x- y/2 y-x/2] Determine if T is projection on a line, reflection in a line, or rotation through an angle, and find the line or angle;
The correct answer is: Projection on the line y = -x
Solve the problem.Let A be a 7 × 9 matrix. Suppose dim Null(A) = 3, find Rank A, Dim Row (A), and Dim Col (A).
The correct answer is: Rank A = 6, Dim Row A = 6, Dim Col A = 6
If .Determine if T is projection on a line, reflection in a line, or rotation through an angle, and find the line or angle. -8x + 6y / 10 6x+8y / 10
The correct answer is: Reflection in the line y = 3x
If .Determine if T is projection on a line, reflection in a line, or rotation through an angle, and find the line or angle.
The correct answer is: T is projection on the line 2x-3y = 0
Find an equation of the line passing through P(0,1,1) and perpendicular to the two lines(d1) [x, y, z] = [1,1,1] + t [0, -1, 2](d2) [x, y, z] = [1,0, -1] + t [2, 1, 1]
The correct answer is: [x, y, z] = [0, 1, 1] +t [-3, 4, 2]
Find all values of a such that {[1, 4, 5], [0, a, 1], [0, 4, a]} is dependent.
The correct answer is: a = 2; a= -2
Determine whether x = (5, 6) lies in U= span{u=(1, 2); v=(0, 1); w=(2, 3)}. If so, write x = a.u+ b.v +c.w then find a+b+c.
The correct answer is: a+b+c = 1
Determine whether x = (-2, -6, -4) lies in U = span{u=(2, 4, 3); v=(1, 1,1)}. If so, write x = a.u+b.v and find a+b.
The correct answer is: a+b=0
Find all values of a and b such that A(a, b, 2) lies on the line x = 1+t; y = 2 - 2t; z = 1-t.
The correct answer is: a=0; b = 4
Find a such that x = (3, 2, a) lies in U = span{(3, 1, 2), (-1, 1, -2), (2, -1, 3)}.
The correct answer is: a=1
Find all values of x such that { [1, 1, 2, [-2, x, 1], [2, -1, 1]} is linearly independent
The correct answer is: all numbers but 3
Which of the following statements are FALSE?(i) The set S = {[-1, 5], [3, -15]}spans R2(ii) The set S = {[-1, 5], [3, -15]}is linearly independent.(iii) The set S = {[-1, 5], [3, -15]}is a basis of R2
The correct answer is: all of them
Given that v1 = [1, -3, 5], v2 =[-3, 8, -2], v3 = [2, -2, 4]. Which of the following statements are true? i) {v1, v2, v3} is linearly independent ii) {v1, v2, v3} is a basis of R3
The correct answer is: both of (i) and (ii)
Find the dimension of the subspace H = { [a+2b+2d, c+d, -3a-6b+4c-2d, -c-d] | a, b, c, d in R}
The correct answer is: dim H = 2
Find the dimensions of the null space and the column space of the given matrix.
The correct answer is: dim Nul A = 3, dim Col A = 2
If A is a 5 × 9 matrix that has rank 2, find dim(Im(A)), dim(Null(A))
The correct answer is: dim(Im(A)) = 2, dim(Null(A)) = 7
Let A be a 20x11 matrix, rank(A) = 8. Find dim(Null(A)); dim(Col(A)).
The correct answer is: dim(Null(A)) = 3; dim(Col(A)) = 8
Let A be a 10x8 matrix, rank(A) = 4. Find dim(row(A)); dim(null(A)).
The correct answer is: dim(row(A) = 4; dim(null(A) = 4
An equation for the plane passing through the points (2, 1, 3), (1, 0, -1) and (4, -2, 0) is
The correct answer is: none of the other choice is true
Given that b1 = [4, 4, -4], b2 = [2, -2, -1] , x = [-2, 10, -1] Determine if x lies in span{b1, b2}. If x lies in B then find u such that x = ub1 +vb2
The correct answer is: x lies B and the coefficient of b1 is 1
Find the equation of the plane passing through the points P(2, 3, 4) and Q(-1, 2, 3) and parallel to the vector w = (3, 4, 5).
The correct answer is: x-12y+9z=2
Find an equation of the plane passing through the point P(1, 0, 1) and containing the line [x, y, z] = [0, 1, -1] +t [2, 0, 1]
The correct answer is: x-3y-2z = -1
Let u = (2; 1; 2) and v = (-1; 0; 1). Find x such that 3u + 7v =|| u || (2x+v)
The correct answer is: x=(2/6; 3/6; 10/6)
Find the parametric equations of the line passing through A(1; 0; -3) and parallel to the line with parametric equations x = -1+t; y = 2-3t; z = 5+t.
The correct answer is: x=1+t; y = -3t; z = -3+t
Find a basis of null A for
The correct answer is: {(-4;-2;1;0;0); (-5;1;0;0;1)}
Let U = {[x, y, z]| x - 2y + z = 0, x + y - z = 0}. Which of the following is a basis for U?
The correct answer is: {[1, 2, 3]}
Let u = (1, 2, 1); v = (2, 3, 0); w = (0, 1, 0). Find projection of u x v on w.
a. (0, 2, 0)
Let P(1,2,1), Q(1,0,-1), R(2,2,0) be the vertices of a parallelogram with adjacent sides RP and RQ. Find the other vertex S.
a. None of the other choices is correct
The volume of the pyramid with vertices (0; 0; 0); (-2; 8; 14); (-6; 7;-3) and (4; 0; 2) is:
b. 70
Which of the vectors below is orthogonal to both (2, 1, -1) and (-3, -2, 4)?
d. (2, -5, -1)
Let u = (3, 3, 6), v = (4, 4, 3), w = (-6, 3, 3) and x = (41, 5, 12). We can writex = au + bv + cw,where a, b, c are numbers. Find b.
d. 2
The angle between the planes x - z = 7 and y - z = 234 is
pi / 3
Let . Find rank(A) and dim(null(A).
rank a = 2 dim (null a) = 2
Let the point P(-1, -1, 2). Find the shortest distance from the point P to the line [x y z]T = [1 1 0]T+t[2 -1 -1]T.
sqrt (28 /3 )
Find the area of the triangle with vertices A(1, 2, 1); B(3; 2, 1), C(0, 5, 2).
sqrt 10
Let u = (1; 1; 1); v = (0; 1; 1) and w = (1; 0; 1). Find the length of x = (3u + v) w.
sqrt 33
Find the area of the triangle with the following vertices A(1, 1, -1), B(2, 0, 1), C(1, -1, 3).
sqrt 5
Let u = (2, 0, 1); v = (3, 1, 0). Find the length of the vector u x (100u+2v).
sqrt 56
Let the point P(2, -1, 0). Find the shortest distance from the point P to the line [x y z]T = [1 1 0]T+t[2 -1 -1]T.
sqrt 7 / sqrt 3
If u = (-2, 1, 1) and v = (1, 0, 1), then || projv (u) || is :
sqrt2 / 2
