1 Systems Of Linear Equations 2 Matrices 3 Determinants 4 Vector Spaces 5 Inner Product Spaces 6 Linear Transformations 7 Eigenvalues And Eigenvectors A Appendix Chapter7: Eigenvalues And Eigenvectors
7.1 Eigenvalues And Eigenvectors 7.2 Diagonalization 7.3 Symmetric Matrices And Orthogonal Diagonalization 7.4 Applications Of Eigenvalues And Eigenvectors 7.CR Review Exercises 7.CM Cumulative Review Section7.CM: Cumulative Review
Problem 1CM Problem 2CM: In Exercises 1 and 2, determine whether the function is a linear transformation. T:M2,2R,... Problem 3CM: Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions... Problem 4CM Problem 5CM: Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4). Problem 6CM: Let T:R4R2 be the linear transformation defined by T(v)=Av, where A=[10100101]. Find a basis for a... Problem 7CM: In Exercises 7-10, find the standard matrix for the linear transformation T. T(x,y)=(3x+2y,2yx) Problem 8CM Problem 9CM Problem 10CM Problem 11CM Problem 12CM Problem 13CM Problem 14CM Problem 15CM Problem 16CM Problem 17CM Problem 18CM Problem 19CM: In Exercises 19-22, find the eigenvalues and the corresponding eigenvectors of the matrix. [7223] Problem 20CM Problem 21CM Problem 22CM Problem 23CM: In Exercises 23 and 24, find a nonsingular matrix P such that P-1AP is diagonal. A=[231012003] Problem 24CM: In Exercises 23 and 24, find a nonsingular matrix P such that P-1AP is diagonal. A=[0354410004] Problem 25CM: Find a basis B for R3 such that the matrix for the linear transformation T:R3R3,... Problem 26CM: Find an orthogonal matrix P such that PTAP diagonalizes the symmetric matrix A=[1331]. Problem 27CM: Use the Gram-Schmidt orthonormalization process to find an orthogonal matrix P such that PTAP... Problem 28CM Problem 29CM Problem 30CM Problem 31CM Problem 32CM: Prove that if A is similar to B and A is diagonalizable, then B is diagonalizable. Problem 1CM
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Transcribed Image Text: Exercises 2.2
1. Suppose that T : R³ → R² is a linear transformation and that
1
-1
3
T
2
and T
2
1
2
3
Compute T | 2
T
and T
3
3
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
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