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TAMU MATH 304 - 4-2

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Math 304 Answers to Selected Problems1 Section 4.25. Find the standard matrix representations for each of the following linearoperators.(a) L is the linear operator that rotates each x in R2by 45◦in theclockwise direction.(b) L is the linear operator that reflects each x in R2about the x1axis and then rotates it 90◦in the counterclockwise direction.(c) L doubles the length of x and then rotates it 30◦in the counter-clockwise direction.(d) L reflects each vector x about the line x2= x1and then projectsit onto the x1axis.Answer: For each of these, we will apply L to the standard basis vec-tors. The resulting vectors are the columns of the matrix representingthe linear operator with respect to the standard basis.(a)L10=1/√2−1/√2L01=1/√21/√2Thus, the matrix is1/√2 −1/√21/√2 1/√2.(b)L10=01L01=101Thus, the matrix is0 11 0.(c)L10=√31L01=−1√3Thus, the matrix is√3 −11√3.(d)L10=00L01=10Thus, the matrix is0 10 0.14. The linear transformation L is defined byL (p(x)) = p0(x) + p(0)maps P3into P2. Find the matrix representation of L with respe ctto the ordered bases [x2, x, 1] and [2, 1 − x]. For each of the followingvectors in p(x) in P3, find the coordinates of L (p(x)) with respect tothe ordered basis [2, 1 − x].(a) x2+ 2x − 3(d) 4x2+ 2xAnswer: First, we find the matrix representing L with respect to thetwo given bases. To do this, we apply L to each of the polynomials inthe first basis:2L(x2) = 2xL(x) = 1L(1) = 1Next, we find the coordinates for each of the above p olynomials in thesecond basis. In this case, we can find the coordinates by inspection.2x = 1(2) − 2(1 − x)1 =12(2) + 0(1 − x)Thus, in basis [2, 1 − x], the polynomial 2x is1−2, and the poly-nomial 1 is.50.Thus, the matrix representation of L with respect to the given matricesis1 .5 .5−2 0 0Now, we can use this matrix in parts (a) and (d). We just find thecoordinates of the given polynomials with respect to the basis [x2, x, 1],and then we multiply by the above matrix.(a)1 .5 .5−2 0 012−3=.5−2(d)1 .5 .5−2 0 0420=5−8315. Let S be the subspace of C[a, b] spanned by ex, xex, and x2ex. Let Dbe the differentiation operator of S. Find the matrix representing Dwith respect to [ex, xex, x2ex].Answer: To find the matrix representing D with respect to this basis,we start by applying D to each basis vector:D(ex) = exD(xex) = xex+ exD(x2ex) = x2ex+ 2xexIn the given basis, excorresponds to the vector100, xex+ excorre-sponds to the vector110, and x2ex+2xexcorresponds to the vector021. Thus, the matrix representing D is1 1 00 1 20 0 118. Let E = [u1, u2, u3] and F = [b1, b2], whereu1= (1, 0, −1)T, u2= (1, 2, 1)T, u3= (−1, 1, 1)Tandb1= (1, −1)T, b2= (2, −1)TFor each of the following linear transformations L from R3into R2, findthe matrix representing L with respect to the ordered bases E and F .(a) L(x) = (x3, x1)T4(c) L(x) = (2x2, −x1)TAnswer:(a) First, we apply L to each of the vectors in the basis E:L(u1) =−11L(u2) =11L(u3) =1−1Next, we change each of these vectors into the basis F. To do this,we find the transition matrix that converts from the standardbasis to F . The transition matrix that converts from F to thestandard basis is S =1 2−1 −1. The transition matrix fromthe standard basis to F is the S−1=−1 −21 1. Thus,−11F=−1 −21 1−11=−1011F=−1 −21 111=−321−1F=−1 −21 11−1=10Thus, the matrix representing L with respect to the bases E andF is−1 −3 10 2 0(c) First, we apply L to each of the vectors in the basis E:5L(u1) =0−1L(u2) =4−1L(u3) =21Next, we change each of these vectors into the basis F by multi-plying by the transition matrix found in part (a):0−1F=−1 −21 10−1=2−14−1F=−1 −21 14−1=−2321F=−1 −21 121=−43Thus, the matrix representing L with respect to the bases E andF is2 −2 −4−1 3


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TAMU MATH 304 - 4-2

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