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Linear transformations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T. Linear transformations Differential equations 1 / 36

Outline 1 Definition of a linear transformation 2 Kernel and range of a linear transformation 3 The eigenvalue-eigenvector problem 4 General results for eigenvalues and eigenvectors Samy T. Linear transformations Differential equations 2 / 36

Definition of mapping Definition 1. Let Then: V and W two vector spaces 1 A mapping is a rule which assigns to each vector v V a vector w = T (v) W 2 Notation: T : V W Samy T. Linear transformations Differential equations 4 / 36

Linear transformation Definition 2. Let V and W two vector spaces A mapping T : V W Then T is a linear transformation if it satisfies: 1 T (u + v) = T (u) + T (v) 2 T (c v) = c T (v) Vocabulary: V is called domain of T W is called codomain of T Samy T. Linear transformations Differential equations 5 / 36

Examples of linear transformations Example with matrices: The map T : M n (R) M n (R), T (A) = A T is a linear transformation Example with functions: The map T : C 2 (I) C 0 (I), T (y) = y + y is a linear transformation Samy T. Linear transformations Differential equations 6 / 36

Matrix form Theorem 3. Let T : R n R m be a linear transformation (e 1,..., e n ) canonical basis of R n Then T is described by the matrix transformation T (x) = A x, where A is the matrix defined by A = [T (e 1 ), T (e 2 ),..., T (e n )] Samy T. Linear transformations Differential equations 7 / 36

Example of matrix form Linear transformation: Defined as T : R 2 R 3 such that T (1, 0) = (2, 3, 1), T (0, 1) = (5, 4, 7) Matrix form: T (x) = A x with A = [T (e 1 ), T (e 2 )] = 2 5 3 4 1 7 General expression: 2x 1 + 5x 2 T (x) = A x = 3x 1 4x 2 x 1 + 7x 2 Samy T. Linear transformations Differential equations 8 / 36

Matrix form for polynomials Linear transformation: Defined as T : P 1 P 2 such that T (a 0 + a 1 x) = (2a 0 + a 1 ) + (a 0 a 1 )x + 3a 1 x 2. Matrix form: If then we have p(x) = a 0 + a 1 x, and a = [ ] a0 a 1 2 1 T (p) = A a, with A = 1 1 0 3 Samy T. Linear transformations Differential equations 9 / 36

Definitions Definition 4. Let T : V W be a linear transformation. Then 1 The kernel of T is defined as a subspace of V : ker(t ) = {v V ; T (v) = 0} 2 The range of T is defined as a subspace of W : Rng(T ) = {w W ; w = T (v) for a v V } Samy T. Linear transformations Differential equations 11 / 36

Example Linear transformation: We set T : C 2 (I) C 0 (I), T (y) = y + y Definition of ker(t ): Solution set of the differential equation y + y = 0 Characterization of ker(t ): We have ker(t ) = { y C 2 (I); y(x) = c 1 cos(x) + c 2 sin(x) } Samy T. Linear transformations Differential equations 12 / 36

Case of a matrix transformation Proposition 5. Let Then T : R n R m be a linear transformation A matrix of T 1 ker(t ) is the solution set to the system Ax = 0 Otherwise stated, we have ker(t ) = nullspace(a) 2 The range of T is characterized as Rng(T ) = colspace(a) Samy T. Linear transformations Differential equations 13 / 36

Application Transformation: We consider T : R 3 R 2 with matrix [ ] 1 2 5 A = 2 4 10 Reduced row-echelon form: For the augmented matrix we find A ker(t ) and Rng(T ): We find [ 1 2 5 ] 0 0 0 0 0 ker(t ) = { x R 3 ; x = r(2, 1, 0) + s(5, 0, 1) } Rng(T ) = { y R 2 ; y = r(1, 2) } dim[ker(t )] + dim[rng(t )] = 3 = dim[r 3 ] Samy T. Linear transformations Differential equations 14 / 36

Rank-nullity theorem Theorem 6. Let V finite dimensional space T : V W linear transformation Then dim[ker(t )] + dim[rng(t )] = dim[v ] Vocabulary: dim[ker(t )] is called nullity of T. Samy T. Linear transformations Differential equations 15 / 36

Example Linear transformation: We define T : P 1 P 2 by T (a + b x) = (2a 3b) + (b 5a)x + (a + b)x 2 Matrix form: In the canonical basis (1, x, x 2 ), 2 3 A = 5 1 1 1 Reduced row-echelon form: For the system Ax = 0, we find 1 0 0 A 0 1 0 0 0 0 Samy T. Linear transformations Differential equations 16 / 36

Example (2) Kernel: According to the reduced row-echelon form, ker(t ) = {0} P 1 Range: According to the reduced row-echelon form, Rng(T ) = span { 2 5x + x 2, 3 + x + x 2} Verifying the rank-nullity theorem: We have dim[ker(t )] = 0, dim[rng(t )] = 2, and thus dim[ker(t )] + dim[rng(t )] = 2 = dim[p 1 ] Samy T. Linear transformations Differential equations 17 / 36

Definitions Definition 7. Let Then A be a n n matrix λ R 1 If the the following system has nontrivial solutions: we say that λ is an eigenvalue of A. A v = λ v, (1) 2 If λ is an eigenvalue A vector v satisfying (1) is called eigenvector. Samy T. Linear transformations Differential equations 19 / 36

Strategy for the eigenvalue problem Illustration in R 2 : Strategy: 1 The eigenvalues are scalars such that det(a λ I) = 0 2 If the eigenvalues λ 1,..., λ k are given by Step 1, The eigenvectors solve the systems (i = 1,..., k) (A λ i ) v i = 0, Samy T. Linear transformations Differential equations 20 / 36

Example with simple eigenvalues Matrix: A = [ ] 5 4 8 7 (2) Characteristic equation: 5 λ 4 8 7 λ λ2 + 2λ 3 = 0 Eigenvalues: λ 1 = 3, λ 2 = 1 Samy T. Linear transformations Differential equations 21 / 36

Example with simple eigenvalues (2) Computation of A λ 1 I: We get Reduced row-echelon form: Eigenvectors for λ 1 = 3: A + 3I = A + 3I [ ] 8 4 8 4 [ ] 1 1 2 0 0 {r v 1 ; r R}, where v 1 = (1, 2) Samy T. Linear transformations Differential equations 22 / 36

Example with simple eigenvalues (3) Eigenvectors for λ 2 = 1: {r v 2 ; r R}, where v 2 = (1, 1) Remark: The family {v 1, v 2 } forms a basis of R 2. Samy T. Linear transformations Differential equations 23 / 36

Example with double eigenvalue Matrix: A = [ ] 1 1 0 1 (3) Characteristic equation: 1 λ 1 0 1 λ (λ 1)2 = 0 Eigenvalues: λ 1 = 1, repeated twice Samy T. Linear transformations Differential equations 24 / 36

Example with double eigenvalues (2) Computation of A λ 1 I: We get directly a reduced row-echelon form A I = [ ] 0 1 0 0 Eigenvectors for λ 1 = 1: We get only one linearly indep. eigenvector {r v 1 ; r R}, where v 1 = (1, 0) Conclusion: Number of indep. eigenvectors is less than order of eigenvalue The matrix A is said to be defective Samy T. Linear transformations Differential equations 25 / 36

Complex eigenvalues Theorem 8. Let A be such that A is an n n matrix A has real-valued elements A admits a complex eigenvalue λ with eigenvector v Then λ is an eigenvalue for A with eigenvector v. Samy T. Linear transformations Differential equations 26 / 36

Example with complex eigenvalues Matrix: A = [ ] 2 6 3 4 Characteristic polynomial: 2 λ 6 p(λ) = 3 4 λ = λ2 2λ + 10 Eigenvalues: λ 1 = 1 + 3ı, λ 2 = λ 1 = 1 3ı Samy T. Linear transformations Differential equations 27 / 36

Example with complex eigenvalues (2) Computation of A λ 1 I: We get A (1 + 3ı)I = Reduced row-echelon form: (A (1 + 3ı)I) Eigenvectors for λ 1 = 1 + 3ı: [ ] 3 3ı 6 3 3 3ı [ 1 1 ı ] 0 0 0 0 {r v 1 ; r R}, where v 1 = ( (1 ı), 1) Samy T. Linear transformations Differential equations 28 / 36

Example with complex eigenvalues (3) Eigenvectors for λ 2 = 1 3ı: {r v 2 ; r R}, where v 2 = v 1 = ( (1 + ı), 1) Remark: The family {v 1, v 2 } forms a basis of C 2. Samy T. Linear transformations Differential equations 29 / 36

David Hilbert Hilbert: Lived 1862-1943 in Germany Among top 3 mathematicians of 20th century Foundations of mathematics Infinite dimensional vector spaces Hilbert spaces Number theory Axioms of geometry Coined the term eigenvalue Eigenvalue: In German, own value (or proper value). Samy T. Linear transformations Differential equations 30 / 36

Eigenspace Definition 9. Let A be a n n matrix λ i eigenvalue of A The eigenspace E i is defined as E i {v; A v = λ i v}. Samy T. Linear transformations Differential equations 32 / 36

Dimension of an eigenspace Theorem 10. Let A be a n n matrix λ i eigenvalue of A m i multiplicity of λ i Then the eigenspace E i is a subspace of C n such that dim[e i ] m i Samy T. Linear transformations Differential equations 33 / 36

Examples Case with simple eigenvalues: For the matrix (2) we have m 1 = dim(e 1 ) = 1, m 2 = dim(e 2 ) = 1 Case with double eigenvalue: For the matrix (3) we have m 1 = 2, dim(e 1 ) = 1 Samy T. Linear transformations Differential equations 34 / 36

Defective and nondefective matrices Definition 11. Let A be a n n matrix. If A has n independent eigenvectors A is called nondefective If A has less than n independent eigenvectors A is called defective Remark: If A is nondefective There exists a basis of eigenvectors of A for R n Samy T. Linear transformations Differential equations 35 / 36

Criterion of nondefectiveness Theorem 12. Examples of nondefective matrices in M n (R): 1 When A has n distinct eigenvalues, A is nondefective 2 A non defective iff for all i we have dim[e i ] = m i Examples: Matrix (2) is nondefective Matrix (3) is defective Samy T. Linear transformations Differential equations 36 / 36