I. Answer each of the following as TRUE or FALSE (answer only). 1. There are linear transformations L: R³ R5 that are onto. 2. If a 3 x 3 matrix A has eigenvalues λ = 1,-1,3, then A is diagonalizable. 3. Let L: V map into 0 W be a linear transformation, the ker L is the set of elements in V which € W. 4. Let L: R³ R4 be a linear transformation, if L is one-to-one, then it is onto. 5. Let L: V → W, if L(O₂) = Ow then L is a linear transformation. 6. If L: V → W is a linear transformation, then for any vector w in W there is a vector v in V such that L(v) = w. 7. If P is nonsingular and D is diagonal, then the eigenvalues of A = P-¹DP are the diagonal entries of D. 8. If is an eigenvalue of A of algebraic multiplicity k, then the dimension of the eigenspace associated with λ is k. 9. A matrix A is diagonalizable if all the roots of its characteristic polynomial are real. 10. Let S = {t² + 1, t-1, t) be a basis for P₂, then [t²-t +3]s =

I. Answer each of the following as TRUE or FALSE (answer only). 1. There are linear transformations L: R³ R5 that are onto. 2. If a 3 x 3 matrix A has eigenvalues λ = 1,-1,3, then A is diagonalizable. 3. Let L: V map into 0 W be a linear transformation, the ker L is the set of elements in V which € W. 4. Let L: R³ R4 be a linear transformation, if L is one-to-one, then it is onto. 5. Let L: V → W, if L(O₂) = Ow then L is a linear transformation. 6. If L: V → W is a linear transformation, then for any vector w in W there is a vector v in V such that L(v) = w. 7. If P is nonsingular and D is diagonal, then the eigenvalues of A = P-¹DP are the diagonal entries of D. 8. If is an eigenvalue of A of algebraic multiplicity k, then the dimension of the eigenspace associated with λ is k. 9. A matrix A is diagonalizable if all the roots of its characteristic polynomial are real. 10. Let S = {t² + 1, t-1, t) be a basis for P₂, then [t²-t +3]s =

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Define T:P2P2 by T(a0+a1x+a2x2)=(2a0+a1a2)+(a1+2a2)xa2x2. Find the eigenvalues and the eigenvectors of T relative to the standard basis {1,x,x2}.
Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
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.
In Exercises 1-12, determine whether T is a linear transformation. T:MnnMnn defines by T(A)=AB, where B is a fixed nn matrix
The gray square in Table 1 has the following vertices: [00],[10],[11],[01] Apply each of the three transformations given in Table 1 to these vertices and sketch the result to verify that each transformation has the indicated effect. Use c=2 in the expansion matrix and c=1 in the shear matrix.
Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.
Let T be a linear transformation from M2,2 into M2,2 such that T([1000])=[1102], T([0100])=[0211], T([0010])=[1201],T([0001])=[3110]. Find T([1314]).