Let R = 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 The column vectors of R represent the homogeneous

Refer to Exercise 1 of Section 1. For each linear transformation L, find the standard matrix representation of L.

For each of the following linear transformations L mapping R3 into R2, find a matrix A such that L(x) = Ax for every x in R3: (a) L((x1, x2, x3)T ) = (x1 + x2, 0)T (b) L((x1, x2, x3)T ) = (x1, x2)T (c) L((x1, x2, x3)T ) = (x2 x1, x3 x2)T

For each of the following linear operators L on R3, find a matrix A such that L(x) = Ax for every x in R3: (a) L((x1, x2, x3)T ) = (x3, x2, x1)T (b) L((x1, x2, x3)T ) = (x1, x1 +x2, x1 +x2 +x3)T (c) L((x1, x2, x3)T ) = (2x3, x2 + 3x1, 2x1 x3)T

Let L be the linear operator on R3 defined by L(x) = 2x1 x2 x3 2x2 x1 x3 2x3 x1 x2 Determine the standard matrix representation A of L, and use A to find L(x) for each of the following vectors x: (a) x = (1, 1, 1)T (b) x = (2, 1, 1)T (c) x = (5, 3, 2)T

Find the standard matrix representation for each of the following linear operators: (a) L is the linear operator that rotates each x in R2 by 45 in the clockwise direction. (b) L is the linear operator that reflects each vector x in R2 about the x1-axis and then rotates it 90 in the counterclockwise direction. (c) L doubles the length of x and then rotates it 30 in the counterclockwise direction. (d) L reflects each vector x about the line x2 = x1 and then projects it onto the x1-axis.

Let b1 = 1 1 0 , b2 = 1 0 1 , b3 = 0 1 1 and let L be the linear transformation from R2 into R3 defined by L(x) = x1b1 + x2b2 + (x1 + x2)b3 Find the matrix A representing L with respect to the bases {e1, e2} and {b1, b2, b3}.

Let y1 = 1 1 1 , y2 = 1 1 0 , y3 = 1 0 0 and let I be the identity operator on R3. (a) Find the coordinates of I(e1), I(e2), and I(e3) with respect to {y1, y2, y3}. (b) Find a matrix A such that Ax is the coordinate vector of x with respect to {y1, y2, y3}.

Let y1, y2, and y3 be defined as in Exercise 7, and let L be the linear operator on R3 defined by L(c1y1 + c2y2 + c3y3) = (c1 + c2 + c3)y1 + (2c1 + c3)y2 (2c2 + c3)y3 (a) Find a matrix representing L with respect to the ordered basis {y1, y2, y3}. (b) For each of the following, write the vector x as a linear combination of y1, y2, and y3 and use the matrix from part (a) to determine L(x): (i) x = (7, 5, 2)T (ii) x = (3, 2, 1)T (iii) x = (1, 2, 3)T

Let R = 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 The column vectors of R represent the homogeneous coordinates of points in the plane. (a) Draw the figure whose vertices correspond to the column vectors of R. What type of figure is it? (b) For each of the following choices of A, sketch the graph of the figure represented by AR and describe geometrically the effect of the linear transformation: (i) A = 1 2 0 0 0 1 2 0 0 0 1 (ii) A = 1 2 1 2 0 1 2 1 2 0 0 0 1 (iii) A = 1 0 2 0 1 3 0 0 1 1

For each of the following linear operators on R2, find the matrix representation of the transformation with respect to the homogeneous coordinate system: (a) The transformation L that rotates each vector by 120 in the counterclockwise direction (b) The transformation L that translates each point 3 units to the left and 5 units up (c) The transformation L that contracts each vector by a factor of one-third (d) The transformation that reflects a vector about the y-axis and then translates it up 2 units 1

Determine the matrix representation of each of the following composite transformations: (a) A yaw of 90, followed by a pitch of 90 (b) A pitch of 90, followed by a yaw of 90 (c) A pitch of 45, followed by a roll of 90 (d) A roll of 90, followed by a pitch of 45 (e) A yaw of 45, followed by a pitch of 90 and then a roll of 45 (f) A roll of 45, followed by a pitch of 90 and then a yaw of 45 1

Let Y , P, and R be the yaw, pitch, and roll matrices given in equations (1), (2), and (3), and let Q = Y PR. (a) Show that Y , P, and R all have determinants equal to 1. (b) The matrix Y represents a yaw with angle u. The inverse transformation should be a yaw with angle u. Show that the matrix representation of the inverse transformation is Y T and that Y T = Y 1. (c) Show that Q is nonsingular, and express Q1 in terms of the transposes of Y , P, and R. 1

Let L be the linear transformation mapping P2 into R2 defined by L(p(x)) = _ 1 0 p(x) dx p(0) Find a matrix A such that L( + x) = A 1

The linear transformation L defined by L(p(x)) = p_ (x) + p(0) maps P3 into P2. Find the matrix representation of L with respect to the ordered bases [x2, x, 1] and [2, 1 x]. For each of the following vectors p(x) in P3, find the coordinates of L(p(x)) with respect to the ordered basis [2, 1 x]: (a) x2 + 2x 3 (b) x2 + 1 (c) 3x (d) 4x2 + 2x 1

Let S be the subspace of C[a, b] spanned by ex , xex , and x2ex . Let D be the differentiation operator of S. Find the matrix representing D with respect to [ex , xex , x2ex ]. 1

Let L be a linear operator on Rn. Suppose that L(x) = 0 for some x _= 0. Let A be the matrix representing L with respect to the standard basis {e1, e2, . . . , en}. Show that A is singular. 1

Let L be a linear operator on a vector space V. Let A be the matrix representing L with respect to the ordered basis {v1, . . . , vn} of V, that is, L(vj ) = #n i=1 ai jvi , j = 1, . . . , n. Show that Am is the matrix representing Lm with respect to {v1, . . . , vn}. 1

Let E = {u1, u2, u3} and F = {b1, b2}, where u1 = 1 0 1 , u2 = 1 2 1 , u3 = 1 1 1 and b1 = (1,1)T , b2 = (2,1)T For each of the following linear transformations L from R3 into R2, find the matrix representing L with respect to the ordered bases E and F: (a) L(x) = (x3, x1)T (b) L(x) = (x1 + x2, x1 x3)T (c) L(x) = (2x2,x1)T 1

Suppose that L1 : V W and L2 : W Z are linear transformations and E, F, and G are ordered bases for V, W, and Z, respectively. Show that, if A represents L1 relative to E and F and B represents L2 relative to F and G, then the matrix C = BA represents L2 L1 : V Z relative to E and G. [Hint: Show that BA[v]E = [(L2 L1)(v)]G for all v V.] 2

Let V and W be vector spaces with ordered bases E and F, respectively. If L : V W is a linear transformation and A is the matrix representing L relative to E and F, show that (a) v ker(L) if and only if [v]E N(A). (b) w L(V) if and only if [w]F is in the column space of A.