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MATH 416 Midterm2
MATH 416 Midterm 2
| Term | Definition |
|---|---|
| Linear maps | A linear map from V to W is a function T: V --> W satisfying additivity and homogeneity [ T(u+v) = Tu + Tv, T λu = λTu ] |
| Zero map | For all u in V, Ou = 0 |
| Identity map / operator | For all u in V, Iu = u |
| Linear map lemma. Suppose v1,...,vn is a basis for V and w1,...wn is a basis for W. | Then there exists a unique linear map T: V --> W such that Tvk = wk for k in (1,...,n). (A linear map is completely determined by its values on a basis.) |
| The sum of linear maps S and T | (S + T) (u) = Su + Tu |
| The product λT | (λT) (u) = λ (Tu) |
| For a linear map T and zero elements 0v and 0w in V and W, | T(0v) = 0w (zero vectors map to zero vectors) |
| Null space (null(T)) | For a linear map T, the null space is the set of vectors mapping to the zero vector. (All u in V such that Tu = 0). null(T) is a subspace of V. |
| Range (range(T)) | For a linear map T, the range is the subset of W consisting of those vectors that are equal to Tu for some u in V. (Tu s.t. u in V). range(T) is a subspace of W. |
| Injectivity (one-to-one) | A function T: V --> W is called injective if T(u) = T(v) implies u = v for all u, v in V. |
| A linear map T is injective iff | A linear map T has null(T) = (0) iff |
| Surjectivity (onto) | A function T: V --> W is called surjective if range(T) = W. |
| Fundamental theorem of linear maps. V is finite-dimensional and T is a linear map. | dim(V) = dim(null(T)) + dim(range(T)) |
| V, W are finite-dimensional vector spaces and T is a linear map from V to W. If dim(V) > dim(W), | Then T is not injective. |
| Matrix representations and vector spaces | The set of matrices with m rows and n columns forms a vector space |
| Linear maps and vector spaces | The set of linear maps from V to W forms a vector space |
| Suppose T is a linear map from V to W, v1,...vn, w1,...wm are bases for V, W. | The matrix T w.r.t those bases is the m-by-n matrix M (T) whose entries Ajk are defined by T(vk) = A1k w1 + ... + Amk wm. This also equals the summation from j=1 to m of Ajk wj |
| M(T) notationally | M (T) = M ( T, v1,...vn, w1,...wn ) |
| Matrix addition | Adds individual entries |
| Scalar multiplication for matrices | Multiples each entry by some λ |
| Matrix addition properties for representations of linear maps. Suppose T and S are linear maps from V to W with fixed bases for V and W. | M (T+S) = M (T) + M (S) |
| Scalar multiplication properties for matrix representations of linear maps. Suppose T is a linear map from V to W with fixed bases for V and W. | M (λT) = λM (T) |
| Matrix multiplication properties for representations of linear maps. Suppose T is a linear map from U to V and S is a linear map from V to W. | M (ST) = M(S) M(T) |
| Matrix multiplication as a summation from m-by-n matrix A and n-by-p matrix B | (AB)jk = r=1,...,n ∑ (Ajr)(Brk) |
| Invertibile, inverse | A linear map T from V to W is invertible if there exists a linear map S from W to V such that ST = Iv and TS = Iw. S is called an inverse of T. |
| If a linear map is invertible, | Then its inverse is unique. We denote this inverse T^-1. |
| A linear map is invertible iff | A linear map is bijective (injective and surjective) iff |
| Suppose V, W are finite-dimensional vector spaces and dim(V) = dim(W) with a linear map T from V to W. Then the following are equivalent: | T is invertible, T is injective, T is surjective |
| Isomorphisms, isomorphic | An isomorphism is an invertible linear map. Two vector spaces are called isomorphic if there is an invertible linear map from one vector space onto the other one. |
| Two finite-dimensional vector spaces over a field F are isomorphic iff | Those two finite-dimensional vector spaces have the same dimension iff. (This implies that any finite-dimensional vector spaces V with dim(V) = n are isomorphic to F^n.) |
| Suppose v1,...vn is a basis of V and w1,...wm is a basis of W and consider the function from linear maps to matrix representations M. | M is an isomorphism between L(V,W) and F^m,n |
| Suppose V and W are finite-dimensional vector spaces and consider L(V,W). | Then L(V,W) is finite-dimensional and dim(L(V,W) = dim(V) dim(W) |
| Invertibility of matrices | A square matrix A in Mn(F) is called invertible if there exists another matrix B in Mn(F) such that AB = In = BA |
| Formula for the inverse of a 2-by-2 matrix | | a b | ---> 1 / (ad-bc) | d -c | | c d | |-b a | |
| Computing the inverse of a matrix | We compute the inverse of a matrix by augmenting it with the identity matrix and row reducing it |
| Let T: V --> W and dim(V) = dim(W). Then T is invertible iff | Let T: V --> W and dim(V) = dim(W). Then M(T, v1,...vn, w1,...,wn) is invertible iff |
| Inverses of bases of the same vector space. Let (a1,...,an) and (b1,...,bn) are bases of V (dim(V) = n). | Then M(Iv, (a1,...,an), (b1,...,bn)) is invertible with the inverse given by M(Iv, (b1,...,bn), (a1,...,an)) |
| Change of basis formula. Suppose T is a linear map from V to V and let (u1,...,un) and (v1,...,vn) be bases of V. Let A = M(T, (u1,...,un), (u1,...,un)), B = M(T, (v1,...,vn), (v1,...,vn), and C = M(T, (u1,...,un), (v1,...,vn)). | A = (C^-1) B C |
| Invariant subspace | Let T be in L(V) and U be a subspace of V. We say U is invariant under T if for all ui in U, T(ui) is in U. |
| Eigenvalues and eigenvectors | Suppose T is in L(V). λ in F is called an eigenvalue of T if there exists a non-zero u in V such that Tu = λu. If this is true, u is called an eigenvector of T corresponding to λ. |
| V has a one-dimensional invariant under T iff | T has an eigenvalue iff |
| Equivalent conditions to be an eigenvalue. Suppose V is finite-dimensional, T is in L(V), and λ is in F. Then the following are equivalent: | a) λ is an eigenvalue of T b) T - λI is not injective c) T - λI is not surjective d) T - λI is not invertible |
| Suppose λ1,...,λm are distinct eigenvalues of T and v1,...,vm are corresponding eigenvectors. | Then v1,...,vm are linearly independent. |
| V, W are finite-dimensional vector spaces and T is a linear map from V to W. If dim(V) < dim(W), | Then T is not surjective |
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