Tema 12: Espacios Vectoriales PDF

# Academia DEIMOS ## Oposiciones: - Secundaria - Diplomados en Estadística del Estado ## Contact: - Phone: 669 31 64 06 - 91 479 23 42 - Website: www.academiadeimos.es - Email: [email protected] - Email: [email protected] # Tema 12: Espacios Vectoriales ## Topics: - Espacios V...

# Academia DEIMOS ## Oposiciones: - Secundaria - Diplomados en Estadística del Estado ## Contact: - Phone: 669 31 64 06 - 91 479 23 42 - Website: www.academiadeimos.es - Email: [email protected] - Email: [email protected] # Tema 12: Espacios Vectoriales ## Topics: - Espacios Vectoriales - Variedades Lineales - Aplicaciones entre Espacios Vectoriales - Teoremas de Isomorfía ## Section 12.1: Concepto de Espacio Vectorial ## Section 12.2: Subespacios Vectoriales ### Section 12.2.1: Definition: - Subspace, or linear variety - A subset of a vector space V that is **itself a vector space with the same operations** - **Equivalently**: For any vectors *u* in U and scalars *λ*, *µ* in K, - *λu* + *µv* is also in U. ### Section 12.2.2: Observations: 1. The zero vector (0) is in every subspace. 2. Both {0} and V are improper subspaces. 3. All other subspaces are proper subspaces. 4. Linear combinations of vectors in U are also in U. ### Section 12.2.3: Definition: - **Subspace spanned by a set** - The set of all linear combinations of vectors in a set S - The smallest subspace that contains S. ### Section 12.2.4: Intersection and Sum of Subspaces: 1. - **Intersection**: The intersection of any family of subspaces is a subspace. - **Union**: The union of subspaces is **not** generally a subspace. 2. - **Sum**: The sum of subspaces is the set of all vectors that can be expressed as sums of vectors from each subspace. - **Direct Sum**: The sum is direct if every vector in the sum has a unique representation as a sum of vectors from each subspace. - This occurs when the intersection of the subspaces is {0}. 3. - **Supplementary Subspaces**: Two subspaces are supplementary if their sum is the entire space and their intersection is {0}. ## Section 12.3: Linear Dependence ### Section 12.3.1: Definition: - A set of vectors is **linearly independent** if the only way to get a linear combination of them that equals 0 is if all the coefficients are 0. - A set of vectors is **linearly dependent** if there exists a linear combination of them that equals 0 where at least one coefficient is not 0. ### Section 12.3.2: Observations: 1. A set of vectors is linearly dependent if and only if one vector can be expressed as a linear combination of the others. 2. A set of vectors is linearly dependent if it contains the zero vector. 3. A non-empty set containing a non-zero vector is linearly independent. 4. Adding vectors to a linearly dependent set makes it linearly dependent. 5. Removing vectors from a linearly independent set makes it linearly independent. ## Section 12.4: Finite Dimensional Spaces ### Section 12.4.1: Definition: - A vector space is **finite dimensional** if it has a finite spanning set (a set of vectors that can be used to generate all vectors in the space) ### Section 12.4.2: Fundamental Theorem of Linear Independence - **Proof**: - Assume that a linearly independent set I has more vectors than a spanning set G. - Express the first vector of I as a linear combination of the vectors in G. - Since I is linearly independent, at least one coefficient in the combination must be non-zero. - Remove that vector from G, producing a smaller spanning set. - Repeat this process for each vector in I until you reach a contradiction. ### Section 12.4.3: Definition: - A **basis** of a finite dimensional vector space is a linearly independent spanning set. ### Section 12.4.4: Theorem: Existence of Bases - **Proof**: Every spanning set contains a basis. ### Section 12.4.5: Theorem: Dimension of a Space - **Proof**: Every basis of a finite dimensional vector space has the same number of vectors. - If there are two bases with differing numbers of vectors, then one must have more vectors than the other. - But by the fundamental theorem, the linearly independent set cannot have more vectors than the spanning set. - Therefore, both bases must have the same number of vectors, which is defined as the **dimension** of the space. ### Section 12.4.6: Consequences: 1. If a spanning set has the same number of vectors as the dimension of the space, then it is a basis. 2. If a linearly independent set has the same number of vectors as the dimension of the space, then it is a basis. ### Section 12.4.7: Theorem: Extension of a Linearly Independent Set - **Proof**: Any linearly independent set can be extended to a basis by adding vectors from another basis until the set has the same number of vectors as the dimension of the space. ### Section 12.4.8: Dimension of a Subspace - **Proof**: The dimension of a subspace is less than or equal to the dimension of the entire vector space. - If the dimension of the subspace is equal to the dimension of the entire vector space, then the subspace is the entire space. ### Section 12.4.9: Coordinates: - If a vector is a linear combination of basis vectors, then the coefficients of the combination are called the **coordinates** of the vector with respect to that basis. ## Section 12.5: Linear Transformations ### Section 12.5.1: Definition: - A **homomorphism**, or **linear transformation**, from a vector space V to a vector space W is a function that preserves vector addition and scalar multiplication. ### Section 12.5.2: Properties: 1. The image of the zero vector in V is the zero vector in W. 2. If a set of vectors is linearly dependent in V, then their images are linearly dependent in W. 3. The composition of two linear transformations is also a linear transformation. ### Section 12.5.3: Determinations of Linear Transformations - A linear transformation from a finite dimensional space V to a space W can be uniquely defined by specifying the images of the basis vectors of V. ### Section 12.5.4: Kernel and Image: - **Kernel**: The set of all vectors in V that map to the zero vector in W. - **Image**: The set of all vectors in W that are the image of some vector in V. ## Section 12.6: Isomorphisms - **Injective (one-to-one):** A linear transformation is injective if and only if its kernel is {0}. - **Surjective (onto):** A linear transformation is surjective if and only if its image is the entire space W. - **Bijective (one-to-one and onto):** A linear transformation is bijective if it is both injective and surjective. - **Isomorphism:** A bijective linear transformation. It preserves the structure of the vector spaces, meaning they are essentially the same. - **Automorphism:** An isomorphism from a vector space to itself. ## Section 12.7: Quotient Vector Space ### Section 12.7.1: Quotient Space: - Given a subspace U of V, a quotient space V/U is formed by considering the equivalence classes of vectors in V where two vectors are equivalent if their difference is in U. - This is denoted as **u + U**, representing the equivalence class of u. - Operations in a quotient vector space: - (u + U) + (v + U) = (u + v) + U - λ(u + U) = λu + U ### Section 12.7.2: Theorem: Dimension of a Quotient Space: - The dimension of the quotient V/U is the difference between the dimension of V and the dimension of U. ### Section 12.7.3: Isomorphism Theorems: 1. **V/ker f is isomorphic to Im f** 2. **(U1 + U2) / U1 is isomorphic to U2 / (U1 ∩ U2)** 3. **(V / U1) / (U2 / U1) is isomorphic to V / U2** ### Section 12.7.4: Consequences 1. **dim(ker f) + dim(Im f) = dim(V)** 2. **Grassmann's Formula**: **dim(U1 + U2) + dim(U1 ∩ U2) = dim(U1) + dim(U2)**