LA 7,8,9

LA 7,8,9

1. Subspaces

   • A subspace of V is a non-empty subset W ⊆ V such that (i) W is closed under addition: u, v ∈ W ⇒ u + v ∈ W; (ii) W is closed under scalar multiplication: v ∈ W, α ∈ K ⇒ αv ∈ W. These two conditions can be replaced with a single condition u, v ∈ W, α, β ∈ K ⇒ αu + βv ∈ W
     • A subspace W is itself a vector space over K under the operations of vector addition and scalar multiplication in V . Notice that all vector space axioms of W hold automatically. (They are inherited from V .)
       • If W1 and W2 are subspaces of V then so is W1 ∩ W2,It is not necessarily true that W1 ∪ W2 is a subspace
       • Let W1, W2 be subspaces of the vector space V . Then W1 + W2 is defined to be the set of vectors v ∈ V such that v = w1+w2 for some w1 ∈ W1, w2 ∈ W2. Or, if you prefer, W1 + W2 = {w1 + w2 | w1 ∈ W1, w2 ∈ W2}
         • If W1, W2 are subspaces of V then so is W1 + W2. In fact, it is the smallest subspace that contains both W1 and W2
         • Let V be a finite-dimensional vector space, and let W1, W2 be subspaces of V . Then dim(W1 + W2) = dim(W1) + dim(W2) − dim(W1 ∩ W2)
           • Let v1, . . . , vn be vectors in a vector space V . The set of all linear combinations α1v1 + α2v2 + · · · + αnvn of v1, . . . , vn forms a subspace of V

2. Linear transformations

   • Let U, V be two vector spaces over the same field K. A linear transformation or linear map T from U to V is a function T : U → V such that (i) T(u1 + u2) = T(u1) + T(u2) for all u1, u2 ∈ U; (ii) T(αu) = αT(u) for all α ∈ K and u ∈ U. Notice that the two conditions for linearity are equivalent to a single condition T(αu1 + βu2) = αT(u1) + βT(u2) for all u1, u2 ∈ U, α, β ∈ K.
   • Let T : U → V be a linear map. Then (i) T(0U ) = 0V ; (ii) T(−u) = −T(u) for all u ∈ U.
     • While many familiar geometrical transformations, such as projections, rotations, reflections and magnifications are linear maps, a nontrivial translation is not a linear map, because it does not satisfy T(0U ) = 0V
       • Linear maps are uniquely determined by their action on a basis). Let U, V be vector spaces over K, let S be a basis of U and let f : S → V be any function assigning to each vector in S an arbitrary element of V . Then there is a unique linear map T : U → V such that for every s ∈ S we have T(s) = f(s)
   • Operations on linear maps
     • (Addition of linear maps). We define a map T1 + T2 : U → V by the rule (T1 + T2)(u) = T1(u) + T2(u) for u ∈ U
     • (Scalar multiplication of linear maps). We define a map αT1 : U → V by the rule (αT1)(u) = αT1(u) for u ∈ U. Now let T1 : U → V and T2 : V → W be two linear maps
     • (Composition of linear maps). We define a map T2T1 : U → W by (T2T1)(u) = T2(T1(u)) for u ∈ U. In particular, we define T 2 = T T and T i+1 = T iT for i > 2
     • Furthermore, for fixed vector spaces U and V over K, the operations of addition and scalar multiplication on the set HomK(U, V ) of all linear maps from U to V makes HomK(U, V ) into a vector space over K
       • Given a vector space U over a field K, the vector space U ∗ = HomK(U, K) plays a special role. It is often called the dual space or the space of covectors of U. One can think of coordinates as elements of U ∗ . Indeed, suppose that U is finite-dimensional and let e1, . . . , en be a basis of U. Every x ∈ U can be uniquely written as x = α1e1 + · · · + αnen, αi ∈ K. The scalars α1, . . . , αn depend on x as well as on the choice of basis, so for each i one can write the coordinate function e i : U → K, e i (x) = αi . It is routine to check that each e i is a linear map, and indeed the functions e 1 , . . . , e n form a basis of the dual space U ∗ .
   • Linear transformations and matrices
     • Let U, V be vector spaces over K of dimensions n, m, respectively. Then, for a given choice of bases of U and V , there is a one-one correspondence between the set HomK(U, V ) of linear maps U → V and the set Km,n of m × n matrices over K.
       • Let T : U → V be a linear map. Let the matrix A = (aij ) represent T with respect to chosen bases of U and V , and let u and v be the column vectors of coordinates of two vectors u ∈ U and v ∈ V , again with respect to the same bases. Then T(u) = v if and only if Au = v
         • 1. Let T1, T2 : U → V be linear maps with m × n matrices A, B respectively. Then the matrix of T1 + T2 is A + B. 2. Let T : U → V be a linear map with m×n matrices A and let λ ∈ K be a scalar. Then the matrix of λT is λA.
           • Let T1 : V → W be a linear map with l × m matrix A = (aij ) and let T2 : U → V be a linear map with m × n matrix B = (bij ). Then the matrix of the composite map T1T2 : U → W is AB.

3. Kernels and Images

   • Let T : U → V be a linear map. The image of T, written as im(T), is the set of vectors v ∈ V such that v = T(u) for some u ∈ U
   • The kernel of T, written as ker(T), is the set of vectors u ∈ U such that T(u) = 0V . i.e. im(T) = {T(u) | u ∈ U}; ker(T) = {u ∈ U | T(u) = 0V }
   • . Let T : U → V be a linear map. Then (i) im(T) is a subspace of V ; (ii) ker(T) is a subspace of U
   • Rank and nullity
     • let T : U → V be a linear map. (i) dim(im(T)) is called the rank of T; (ii) dim(ker(T)) is called the nullity of T
       • The rank-nullity theorem). Let U, V be vector spaces over K with U finite-dimensional, and let T : U → V be a linear map. Then rank(T) + nullity(T) = dim(U)
       • Let T : U → V be a linear map, and suppose that dim(U) = dim(V ) = n. Then the following properties of T are equivalent: (i) T is surjective; (ii) rank(T) = n; (iii) nullity(T) = 0; (iv) T is injective; (v) T is bijective;
     • Rank of a Matrix
       • Let T : U → V be a linear transformation, and let e1, . . . , en be a basis of U. Then the rank of T is equal to the size of the largest linearly independent subset of T(e1), . . . , T(en).
         • 1. The row space of A is the subspace of Kn spanned by the rows r1, . . . , rm of A. The row rank of A is equal to the dimension of the row space of A. . 2. The column space of A is the subspace of Km,1 spanned by the columns c1, . . . , cn of A. The column rank of A is equal to the dimension of the column space of A.
           • Suppose that the linear map T has matrix A. Then rank(T) is equal to the column rank of A.
             • Applying elementary row operations (R1), (R2) or (R3) to a matrix does not change the row or column rank. The same is true for elementary column operations (C1), (C2) and (C3).
               • Let s be the number of non-zero rows in the row and column reduced form of a matrix A (see Theorem 3.6). Then both row rank of A and column rank of A are equal to s

4. More Kernels and Images

   • The rank of a matrix A is equal to the number of non-zero rows after reducing A to upper echelon form.
   • The rank criterion
     • Let A be the augmented n × (m + 1) matrix of a linear system. Let B be the n × m matrix obtained from A by removing the last column. The system of linear equations has a solution if and only if rank(A) = rank(B).
     • Solving Simulataneous Equations Revisited
       • In general, it is easy to see (and you should work out the details) that if x is one solution to a system of equations, then the complete set of solutions is equal to x + nullspace(A) = {x + y | y ∈ nullspace(A)}.