Coordinates. Definition Let B = { v 1, v 2,..., v n } be a basis for a vector space V. Let v be a vector in V, and write

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1 MATH10212 Linear Algebra Brief lecture notes 64 Coordinates Theorem 6.5 Let V be a vector space and let B be a basis for V. For every vector v in V, there is exactly one way to write v as a linear combination of the vectors in B. Let B = { v 1, v 2,..., v n } be a basis for a vector space V. Let v be a vector in V, and write v = c 1 v 1 + c 2 v c n v n Then c 1, c 2,..., c n are called the coordinates of v with respect to B, and the column vector c 1 c 2 [ v] B =. is called the coordinate vector of v with respect to B. c n Theorem 6.6 Let B = { v 1, v 2,..., v n } be a basis for a vector space V. Let u and v be vectors in V and let c be a scalar. Then a. [ u + v] B = [ u] B + [ v] B b. [c u] B = c[ u] B Theorem 6.7 Let B = { v 1, v 2,..., v n } be a basis for a vector space V and let u 1,..., u k be vectors in V. Then { u 1,..., u k } is linearly independent in V if and only if {[ u 1 ] B,..., [ u k ] B } is linearly independent in R n.

2 MATH10212 Linear Algebra Brief lecture notes 65 Dimension Theorem 6.8 Let B = { v 1, v 2,..., v n } be a basis for a vector space V. a. Any set of more than n vectors in V must be linearly dependent. b. Any set of fewer than n vectors in V cannot span V. Theorem 6.9 The Basis Theorem If a vector space V has a basis with n vectors, then every basis for V has exactly n vectors. A vector space V is called finite-dimensional if it has a basis consisting of finitely many vectors. The dimension of V, denoted by dimv, is the number of vectors in a basis for V. The dimension of the zero vector space { 0} is defined to be zero. A vector space that has no finite basis is called infinite-dimensional. Theorem 6.10 Let V be a vector space with dim V = n. Then a. Any linearly independent set in V contains at most n vectors. b. Any spanning set for V contains at least n vectors. c. Any linearly independent set of exactly n vectors in V is a basis for V. d. Any spanning set for V consisting of exactly n vectors is a basis for V. e. Any linearly independent set in V can be extended to a basis for V. f. Any spanning set for V can be reduced to a basis for V. Theorem 6.11 Let W be a subspace of a finite-dimensional vector space V. Then a. W is finite-dimensional and dimw dimv. b. dimw =dimv if and only if W = V.

3 MATH10212 Linear Algebra Brief lecture notes 66 Change of Basis Change-of-Basis Matrices Let B = { u 1,..., u n } and C = { v 1,..., v n } be bases for a vector space V. The n n matrix whose columns are the coordinate vectors [ u 1 ] C,..., [ u n ] C of the vectors in B with respect to C is denoted by P C B and is called the change-of-basis matrix from B to C. That is P C B = [[ u 1 ] C [ u 2 ] C... [ u n ] C ] Theorem 6.12 and Let be bases for a vector space V and let B = { u 1,..., u n } C = { v 1,..., v n } P C B be the change-of-basis matrix from B to C. Then a. P C B [ x] B = [ x] C for all x in V. b. P C B is the unique matrix P with the property that P [ x] B = [ x] C for all x in V. c. P C B is invertible and (P C B ) 1 = P B C. Linear Transformations A linear transformation from a vector space V to a vector space W is a mapping such that, for all u and v in V and all scalars c, 1. T ( u + v) = T ( u) + T ( v) 2. T (c u) = ct ( u) It is straightforward to show that this definition is equivalent to the requirement that T preserve all linear combinations. That is, is a linear transformation if and only if T (c 1 v c k v k ) = c 1 T ( v 1 ) + + c k T ( v k ) for all v 1,..., v k in V and scalars c 1,..., c k. Properties of Linear Transformations

4 MATH10212 Linear Algebra Brief lecture notes 67 Theorem 6.14 Let be a linear transformation. Then a. T ( 0) = 0) b. T ( v) = T ( v) for all v in V. c. T ( u v) = T ( u) T ( v) for all u and v in V. Theorem 6.15 Let be a linear transformation and let B = { v 1,..., v n } be a spanning set for V. Then T (B) = {T ( v 1 ),..., T ( v n )} spans the range of T. Composition of Linear Transformations and If T : U V S : V W are linear transformations, then the composition of S with T is the mapping S T, defined by (S T )( u) = S(T ( u)) where u is in U. Theorem If T : U V and S : V W are linear transformations, then S T : U W is a linear transformation. Inverses of Linear Transformations A linear transformation is invertible if there is a linear transformation T : W V such that T T = I V and T T = I W In this case, T is called an inverse for T.

5 MATH10212 Linear Algebra Brief lecture notes 68 Theorem 6.17 is unique. If T is an invertible linear transformation, then its inverse The Kernel and Range of a Linear Transformation Let be a linear transformation. The kernel of T, denoted ker(t ), is the set of all vectors in V that are mapped by T to 0 in W. That is, ker (T ) = { v in V : T ( v) = 0} The range of T, denoted range(t ), is the set of all vectors in W that are images of vectors in V under T. That is, range (T ) = {T ( v) : v in V } = { w in W : w = T ( v) for some v in V } Theorem Let be a linear transformation. Then a. The kernel of T is a subspace of V. b. The range of T is a subspace of W. Let be a linear transformation. The rank of T is the dimension of the range of T and is denoted by rank(t ). The nullity of T is the dimension of the kernel of T and is denoted by nullity(t ). Theorem The Rank Theorem Let be a linear transformation from a finite-dimensional vector space V into a vector space W. Then rank (T ) + nullity (T ) = dim (T ) One-to-One and Onto Linear Transformations A linear transformation is called one-to-one if T maps distinct vectors in V to distinct vectors in W. If range(t ) = W, then T is called onto.

6 MATH10212 Linear Algebra Brief lecture notes 69 Theorem A linear transformation is one-to-one if and only if ker(t ) = { 0}. Theorem Let dimv = dimw = n. Then a linear transformation is one-to-one if and only if it is onto. Theorem Let be a one-to-one linear transformation. If S = { v 1,..., v k } is a linearly independent set in V, then T (S) = {T ( v 1 ),..., T ( v k )} is a linearly independent set in W. Corollary Let dimv = dimw = n. Then a one-to-one linear transformation maps a basis for V to a basis for W. Theorem A linear transformation is invertible if and only if it is one-to-one and onto. Isomorphisms of Vector Spaces A linear transformation is called an isomorphism if it is one-to-one and onto. If V and W are two vector spaces such that there is an isomorphism from V to W, then we say that V is isomorphic to W and write V W. Theorem 6.25 Let V and W be two finite-dimensional vector spaces. Then V is isomorphic to W if and only if dimv = dimw. The Matrix of a Linear Transformation Theorem 6.26 Let V and W be two finite-dimensional vector spaces with bases B and C, respectively, where B = { v 1,..., v n }. If is a linear transformation, then the m n matrix A defined by A = [[T ( v 1 )] C [T ( v 2 )] C [T ( v n )] C ]

7 MATH10212 Linear Algebra Brief lecture notes 70 satisfies for every vector v in V. A[ v] B = [T ( v)] C Matrices of Composite and Inverse Linear Transformations Theorem Let U, V and W be finite-dimensional vector spaces with bases B, C, and D, respectively. Let T : U V and S : V W be linear transformations. Then [S T ] D B = [S] D C [T ] C B Theorem Let be a linear transformation between n- dimensional vector spaces V and W and let B and C be bases for V and W, respectively. Then T is invertible if and only if the matrix [T ] C B is invertible. In this case, ([T ] C B ) 1 = [T 1 ] B C Change of Basis and Similarity Theorem Let V be a finite-dimensional vector space with bases B and C and let T : V V be a linear transformation. Then [T ] C = P 1 [T ] B P where P is the change-of-basis matrix from C to B. Let V be a finite-dimensional vector space and let T : V V be a linear transformation. Then T is called diagonalizable if there is a basis C for V such that matrix [T ] C is a diagonal matrix. Theorem Version 4 Let A be an n n matrix and let The Fundamental Theorem of Invertible Matrices: be a linear transformation whose matrix [T ] C B with respect to bases B and C of V and W, respectively, is A. The following statements are equivalent:

8 MATH10212 Linear Algebra Brief lecture notes 71 a. A is invertible. b. A x = b has a unique solution for every b in R n. c. A x = 0 has only the trivial solution. d. The reduced row echelon form of A is I n. e. A is a product of elementary matrices. f. rank(a) = n g. nullity(a) = 0 h. The column vectors of A are linearly independent. i. The column vectors of A span R n. j. The column vectors of A form a basis for R n. k. The row vectors of A are linearly independent. l. The row vectors of A span R n. m. The row vectors of A form a basis for R n. n. det A 0 o. 0 is not an eigenvalue of A. p. T is invertible. q. T is one-to-one. r. T is onto. s. ker(t ) = { 0} t. range(t ) = W

9 MATH10212 Linear Algebra Brief lecture notes 72 Short-list of theoretical (bookwork) questions Ideally, you should know the proof of every theorem in the module: there is certainly nothing extra in it, everything belongs to the basics, bare necessities. But to make it easier for you to prepare for the exam, here is a short-list of theoretical questions, some of which will occur in the exam paper. At the exam, you do not have to reproduce the proofs in the lectures word-by-word. Common sense rules apply: if you are asked to prove something, you cannot just say...because it was proved in the lectures ; on the other hand, there is no need to prove previous lemmas and theorems on which the proof of the required bit was based in the lectures. 1. All the definitions and statements of theorems, lemmas, etc. 2. Explain why it is legitimate to use elementary row operations of the augmented matrix for solving a system of linear equations. 3. Prove that a finite system of vectors is linearly dependent if and only if (at least) one of them is a linear combination of the others. 4. Prove that e.r.o.s do not alter the span of the rows of a matrix. 5. Prove that (AB) T = B T A T. 6. Prove that A + A T and AA T are symmetric matrices. 7. Prove that if A is an invertible matrix, then a linear system A x = b has a unique solution. 8. Prove that (AB) 1 = B 1 A 1 if A 1 and B 1 exist. 9. Prove that if the r.r.e.f. of a square matrix A is I, then A is a product of elementary matrices. 10. Prove that if a square matrix A is a product of elementary matrices, then A is invertible. 11. Prove that a right or left inverse of a square matrix is a genuine two-sided inverse of it. 12. Explain why the Gauss Jordan double matrix method works for finding the inverse matrix. 13. Prove that the solution set of any homogeneous system A m n x = 0 is a subspace of R n. 14. Prove the Rank Nullity Theorem for matrices. 15. Suppose that vectors e 1,..., e k form a basis of a vector (sub)space U. Prove that then any vector in U can be uniquely represented as a linear combination of the e i. 16. Prove that every linear transformation is a matrix transformation.

10 MATH10212 Linear Algebra Brief lecture notes Prove that if a matrix is orthogonally diagonalizable, then it is symmetric. 18. Prove that the set E λ of all eigenvectors of an n n matrix A corresponding to an eigenvalue λ together with the zero vector is a subspace of R n. 19. Prove that the product of two orthogonal matrices (of the same size) is also an orthogonal matrix, and that the inverse of an orthogonal matrix is an orthogonal matrix. 20. Prove that eigenvectors for pairwise different eigenvalues are linearly independent. 21. Prove that similar matrices have the same eigenvalues, the same rank, and the same determinant. 22. Prove that det(ab) = det A det B. 23. Prove that the determinant of an orthogonal matrix is 1 or Prove that eigenvectors of a symmetric matrix corresponding to different eigenvalues are orthogonal. 25. Prove that the kernel of a linear transformation is a subspace of V, and the range of T is a subspace of W. 26. Prove that a linear transformation is one-to-one if and only if ker(t ) = { 0}.