Let V, W be finite dimensional vector spaces and T: V → W be a linear transformation. Let A = [T]½ and B= [T], where B, B' and Y, Y' are bases of V and W respectively. Let R be the RREF of A and R' be the RREF of B. (a) Prove that the number of leading ones in R is equal to the number of leading ones in R'. (b) Show by example, that R and R' need not be equal.

Let V, W be finite dimensional vector spaces and T: V → W be a linear transformation. Let A = [T]½ and B= [T], where B, B' and Y, Y' are bases of V and W respectively. Let R be the RREF of A and R' be the RREF of B. (a) Prove that the number of leading ones in R is equal to the number of leading ones in R'. (b) Show by example, that R and R' need not be equal.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 24EQ

See similar textbooks

Similar questions

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 the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
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)
Find a basis for R2 that includes the vector (2,2).
Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.
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 v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.
Prove that in a given vector space V, the additive inverse of a vector is unique.
Prove that in a given vector space V, the zero vector is unique.
Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions of Rn and Rm.
Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}