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For problem 9-15, determine
(a) Computing
(b) Direct calculation.
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Differential Equations and Linear Algebra (4th Edition)
- Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).arrow_forwardLet T:P2P4 be the linear transformation T(p)=x2p. Find the matrix for T relative to the bases B={1,x,x2} and B={1,x,x2,x3,x4}.arrow_forwardLet T be a linear transformation from R2 into R2 such that T(1,0)=(1,1) and T(0,1)=(1,1). Find T(1,4) and T(2,1).arrow_forward
- Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1),T(v2),T(v3)} is linearly dependent.arrow_forward2. Now let W be the line y = 2x 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 as before. (c) Use that matrix to calculate T [4] -1 (d) Draw the vector [4] the line W, and the projection T[1] -1 -1arrow_forwardLet A = 10-3 -3 16 2-2 -1 -2 3 1 Consider the transformation T defined by T(x)=Ax. Find a vector x whose image under T is vector b. Analyze whether x is unique. Describe the arguments on which you base your answers.arrow_forward
- Solve the problem. Let T: ²² be a linear transformation that maps u = · [1] into [3] Use the fact that T is linear to find the image of 3u+v. -8 28 -28 42 [] -36 [] 14 0 [11] -12 and maps v = · =[3] into [4]. ·arrow_forward[-2 3 1 , and define the transformation T : R° → R² by T(x) = Ax for each x E R°. For this transformation: -5 Let the matrix A = -3 4 2 (a): For the vector u = 1 find the image under T of u. -1 8 , and explain why this vector makes sense and has this image. -12 (b): Find a vector w whose image under T isarrow_forwardPlease help with the following linear algebra, linear transformation problem. Please use as much detail as possible. I've looked at sources in how to do this problem but none of them make sense to me. Thanks in advance!arrow_forward
- 5. Find the standard matrix of the given linear transformation: (a) P: R? - R? projects a vector onto the line y = -2x. For the vector u = find P(ü). (b) Rg: R3 - R rotates a vector clockwise about the z - axis through the angle 30°. For the 21 vector u = 2| find R9 (ũ). 6. For the following linear transformations find S • T and T • S. Indicate the domain and codomain of the new linear transformations. x+ y+z] x-y-zl 2x + 7. Find the inverse of the following linear transformation: [3x + 2y] [5x + 3yl %3Darrow_forwardLet T: R² R2 be a linear transformation that sends the vector u = (5,2) into (2, 1) and maps V = = (1, 3) into (−1,3). Use properties of a linear transformation to calculate the following. (Enter your answers as ordered pairs, such as (1,2), including the parentheses.) T(-4u) = = T(8v) = T(-4u +8v) =arrow_forwardLet 4-[1] and W-[*] A = w= -[-1³]. Find k so that there exists a vector X whose image under the linear transformation T(x) = Axis w. Note: The image is what comes out of the transformation. k = Find k so that w is a solution of the equation Ax = 0. k =arrow_forward
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