Vector Spaces and Subspaces

Transcription

1 Vector Spaces and Subspaces Our investigation of solutions to systems of linear equations has illustrated the importance of the concept of a vector in a Euclidean space. We take time now to explore the formal structure of these sets of vectors. Euclidean n-space R n is the set of all vectors of the form (x 1, x 2,,x n ) (written here as row vectors, but equally well-defined as column vectors). For a fixed choice of n, the set R n of vectors comes equipped with an arithmetic we have studied that includes an addition of vectors and a scalar multiplication. This arithmetic can be formalized as follows. Any set of vectors V on which there are defined two operations, addition (symbolized as +) and scalar multiplication (symbolized by, or more simply by concatenation of the symbols of the scalar and vector), that satisfy the conditions listed below is called a vector space. The necessary conditions for vector addition are: 1. Vector addition is closed within the vector space: given u,v V, the sum u + v also lies in V. 2. Vector addition is commutative: for all vectors u,v V, u + v = v + u. 3. Vector addition is associative: for all vectors u,v,w V, u + (v + w) = (u + v) + w.

2 4. There is a zero vector in V, a vector denoted 0 that acts as an additive identity: v + 0 = v = 0 + v for any v V. 5. Every vector v V possesses an additive inverse in V, a vector denoted v for which v + ( v) = 0. The necessary conditions for scalar multiplication are: 6. Scalar multiplication is closed within V: given a scalar c and vector v V, the scalar product cv is a vector in V. 7. Scalar multiplication distributes over vector addition: given scalars c, d and vectors u,v V, it is always true that c(u + v) = cu + cv and (c + d)v = cv + dv. 8. Scalar multiplication is associative: given scalars c, d and any vector v V, we have (cd)v = c(dv). 9. The scalar 1 is a multiplicative identity: for any v V, 1v = v. It should be clear by now that for any integer n 0, Euclidean n-space R n meets all the criteria for being a vector space. It is significant, however, that other sets of objects different from R n are also vector spaces: 1. The set of complex numbers: C = { x +iy x, y R } 2. The set of all m n matrices: R m n

3 3. The trivial vector space {0}, consisting of the single vector 0 4. The set of all sequences of real numbers R N = {(x 1,x 2, ) x i R } 5. The set of all doubly-infinite sequences of real numbers R Z = {(,x 1, x 0, x 1, x 2, ) x i R } 6. The set of all polynomials of degree n in one variable with real coefficients: P n = { p(t ) = a 0 +a 1 t + a 2 t a n t n a i R } 7. Polynomials (of arbitrary degree): R[t ] = { p(t ) = a 0 + a 1 t + + a n t n a i R,a n 0,n 0} 8. The set of formal power series in the variable t: R[[t ]] = { p(t ) = a 0 +a 1 t +a 2 t 2 + a i R} (the term formal means that convergence issues are of no interest, as opposed to Taylor series, for which the chief question of concern is convergence) 9. The set of rational expressions (quotients of polynomials) in the variable t: R(t ) = { p(t ) / q(t ) p(t ),q(t ) R[t ]} 10. The set of Laurent series in t: R((t )) = {a m t m + a m+1 t m+1 + a i R,m Z,a m 0 } (the initial index in a Laurent series can be negative!)

4 11. The set R D = {f : D R } of all real-valued functions on some domain D (under pointwise addition); the most important such function space corresponds to D = R All these vector spaces share the feature that the set of scalars used in vector arithmetic is the set of real numbers. We call R the scalar field for these vector spaces, which are, as a result, called real vector fields. Another important scalar field is the complex numbers. Vector spaces that use complex numbers as scalars are complex vector spaces. Here are some examples: 1. Complex n-space: C n = {(z 1,z 2,,z n ) z i C} 2. The set of all m n complex matrices: C m n 3. The trivial vector space {0} 4. Polynomials with complex coefficients: C[t ] = { p(t ) = a 0 + a 1 t + + a n t n a i C,a n 0,n 0} 5. Complex formal power series in the variable t: C[[t ]] = { p(t ) = a 0 + a 1 t + a 2 t 2 + a i C} 6. The set C D = { f :D C} of all real-valued functions on some domain D

5 Subsets of vector spaces also have the structure of a vector space. For this reason, we define a subspace of a vector space V to be any nonempty subset H that contains the zero vector of V and under which the vector addition and scalar multiplication from V are closed. For instance, the subset {0} of R 3 consisting of only the zero vector is a subspace of R 3, called the zero subspace or the trivial subspace. Also, the subset {c(1,1, 1) c a scalar} of all scalar multiples of the fixed vector (1,1, 1) is a subspace of R 3, as is the subset {(x 1,0, x 3 )} consisting of all vectors that lie in the x 1 x 3 -plane. Note that in each of these cases the subspace can all be characterized as the span of a small number of fixed vectors: {0 } = Span{0} {c(1,1, 1)} = Span{(1,1, 1)} {(x 1,0, x 3 )} = Span{(1,0,0),(0,0,1)} This is no coincidence. One part of establishing the general truth of this statement is contained in the Theorem If the vector space V contains the vectors v 1,v 2,,v p, then their span is a subspace of V.

6 Proof The span of a finite number of vectors consisits of all possible linear combinations of these vectors. In particular, it includes the trivial combination 0v 1 + 0v 2 + 0v p = 0. Also, it is clear that the sum of any two linear combinations of v 1,v 2,,v p is also a linear combination of v 1,v 2,,v p, as is any scalar multiple of a linear combination of these vectors. Thus, their span is a subspace of V. // The proof that every subspace of R n is the span of a finite and fixed number of vectors is a bit more challenging, and will be the goal of our discussion as we move on. Observe also that the three subspaces of R 3 in our example above could be identified as, in turn, a single point (the trivial subspace), a line ( Span{(1,1, 1)}) and a plane ( {(x 1,0, x 3 )}). Once we have established that every subspace of R n is the span of a finite and fixed number of vectors, it will follow that every such subspace is geometrically a hyperplane in R n passing through the origin (i.e., containing the zero vector). It is for this reason that we conventionally use the symbol H for subspace. Note also, however, that no point, line or plane in R 3 is a subspace unless it contains 0.