3. Let U, V, W be vector spaces, and let T₁: V→W and T2: W U be linear transformations. Prove that TV U defined by T = T20 T₁ is a linear transformation.
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- Let 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).Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions of Rn and Rm.
- Let T be a linear transformation from R3 into R such that T(1,1,1)=1, T(1,1,0)=2 and T(1,0,0)=3. Find T(0,1,1)Let T:R3R3 be the linear transformation that projects u onto v=(2,1,1). (a) Find the rank and nullity of T. (b) Find a basis for the kernel of T.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.
- Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors {vu,wv,uw} is linearly independent or linearly dependent.Find a basis for R2 that includes the vector (2,2).Let T be a linear transformation from R2 into R2 such that T(x,y)=(xcosysin,xsin+ycos). Find a T(4,4) for =45, b T(4,4) for =30, and c T(5,0) for =120.
- For the linear transformation from Exercise 45, let =45 and find the preimage of v=(1,1). 45. Let T be a linear transformation from R2 into R2 such that T(x,y)=(xcosysin,xsin+ycos). Find a T(4,4) for =45, b T(4,4) for =30, and c T(5,0) for =120.Show that T from Exercise 71 is represented by the matrix A=[12121212]. Proof Let T be the function that maps R2 into R2 such that T(u)=projvu, where v=(1,1) (a) Find T(x,y) (b) Find T(5,0) (c) Prove that T is a linear transformation from R2 into R2.Let V be an inner product space. For a fixed nonzero vector v0 in V, let T:VR be the linear transformation T(v)=v,v0. Find the kernel, range, rank, and nullity of T.