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Problems
For problem 9-15, determine
(a) Computing
(b) Direct calculation.
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Chapter 6 Solutions
EBK DIFFERENTIAL EQUATIONS AND LINEAR A
- Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).arrow_forwardThe cross product of two vectors in R° is defined by a1 [ azbz – azb2 a2 b2 azbı – a,b3 az b3 [a,b2 – azbı . - 3 Let v = 3 Find the matrix A of the linear transformation from R° to R given by T(7) = i xã. A =arrow_forward(a) Let u and v be (fixed, but unknown) vectors in R". Suppose that T: R" → R" is a linear transformation such that T(u) = 6u + v and T(v) = 4u - 2v. Compute (T. T)(v), where TT is the composition of T with itself. Express your answer as a linear combination of u and v. (ToT)(v) = 10 u + -1 V Incorrect answer. Incorrect answer. (b) Let v and w be (fixed, but unknown) vectors in R", which are not scalar multiples of each others. Suppose that T: R" → R" is a linear transformation such that T(4v+3w) = -2v-2w and T(v+1w) = 5v+2w. Compute T(v) and express it as a linear combination of v and w. T(v) = 4 v + 2 W Incorrect answer. Incorrect answer.arrow_forward
- Q.3. Let R, (x) be a reflection of any vector x through the line along the vector u and P, (x) be a projection of any vector x onto the line along the vector u. Find the transformation RP for u =arrow_forwardSuppose T: R³-R² is a linear transformation. Let u and y be the vectors given below, and suppose that T(u) and T(v) are as given. Find T(-u-3v). -5 V = -2 -3 5 u = 3 2 0 T(-u-3v) = 0 T(u) = [] T(v) = 0 4arrow_forward1. Let W be the line y = x in R². (a) Find a unit vector u on W. (b) Consider the linear transformation T: R² R² that projects each vector orthogonally onto W. Calculate the matrix for T which is uu¹. (c) Use that matrix to calculate T (d) Draw the vector [] [4] T[A] the line W, and the projection Tarrow_forward
- Consider the following. Tis the reflection through the origin in R2: T(x, y) = (-x, -y), v = (3, 4). (a) Find the standard matrix A for the linear transformation T. (b) Use A to find the image of the vector v T(v) = (c) Sketch the graph of v and its image. T (V) -5 -4 -3 -2 -1 1 2 3 45 -2 T(V) T(v) -5 -4 -3 -2 -1 T(v)arrow_forwardSuppose T: R3-R2 is a linear transformation. Let u, v and w be the vectors given below, and suppose that T(u) and T(v) are as given. Find T(w). u = 122 -2 V = 2 -5 4 -3 w=-10 8 T(u) = T(v) = 5 T(w)arrow_forwardGiven the linear transformation T: x+ T(x) = A x where -2 -2 A = 2 -1 Find all vectors x E R2 where T(x)=0.arrow_forward
- Asaparrow_forwardLet T : R³ → R³ be the linear transformation that does the following things, in this order, to an input vector x = [x y z]¹: (i) Interchanges the second and third coordinates of . (ii) Multiplies the first coordinate of the resulting vector by 2. (iii) Replaces the second coordinate of the resulting vector with a 0. (iv) Multiplies the resulting vector by the following matrix: 0 0 0 0 You don't have to show that T is linear. (a) The description of T given above is purely algebraic, in that it explicitly describes how to take 7 = [x y z] and write down T() in coordinates. Give a geometric description of what each of the four "steps" of applying T actually does to a vector. (Your Week 9 tutorials may help in describing what the last step does.) (b) Find the standard matrix AT of T. (c) Find a spanning set for null(AT), and describe what null(AT) is geometrically (i.e., describe it geomet- rically as a subset of R³) (d) Find a spanning set for im(AT), and describe what im(AȚ) is…arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
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