Vector space and subspace Math 112, week 8 Goals: Vector space, subspace. Linear combination and span. Kernel and range (null space and column space). Suggested Textbook Readings: Sections 4.1, 4.2
Week 8: Vector space and subspace 2 Definition (Vector space): (V, +, ) A vector space is a non-empty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the ten axioms listed below: 1. The sum of u, v, denoted by u + v, is in V. 2. The scalar multiple c u is in V. 3. u + v = v + u. 4. ( u + v) + w = u + ( v + w). 5. There is a zero vector 0 in V such that u + 0 = u. 6. For each u in V, there is a vector u in V such that u + ( u) = 0. 7. c( u + v) = c u + c v. 8. (c + d) u = c u + d u. 9. c(d u) = (cd) u. 10. 1 u = u. The spaces (R n, +, ) are premier examples of vector spaces.
Week 8: Vector space and subspace 3 Examples of vector spaces. 1. Let V = F(R; R) be the set of all functions f : R R, i.e. the set of all real-valued functions of a real viable. We define the addition + : the scalar multiplication : 2. Let P n be the set of all polynomials of degree at most n: p(t) = a 0 + a 1 t + a 2 t 2 + + a n t n If q(t) = b 0 + b 1 t + b 2 t 2 + + b n t n is another element in P n, define The addition + : The scalar multiplication :
Week 8: Vector space and subspace 4 3. Let M 2 2 denote the set of all 2 2 matrices. We define on M 2 2 the addition + and scalar multiplication in the usual way. Then (M 2 2, +, ) is a vector space. Example that is not a vector space: 1. On the set R 2, define an addition operation, denote by + as follows: x + u = x + v y v y + u and define the scalar multiplication as usual. Then (R 2, +, ) is not a vector space.
Week 8: Vector space and subspace 5 Definition (Subspace) A subspace of a vector space (V, +, ) is a subset H of V such that (H, +, ) itself is also a vector space. Equivalently, this means Example 1: Determine if H is a subspace of V. 1. Zero subspace: The set H = { 0} is a subspace of V. 2. H = P n and V = F(R; R). 3. The set s H = { t : s and t are real} 0 and V = R 3.
Week 8: Vector space and subspace 6 4. Is R 2 a subspace of R 3? 5. Let H = { x : x + y = 1}, is H a subspace of R 2? y 6. H = { x : xy 0}, is H a subspace of R 2? y Linear combination and span.
Week 8: Vector space and subspace 7 Example 2: Given v 1 and v 2 in a vector space V, let H = Span{ v 1, v 2 }. Show that H is a subspace of V. Theorem: If v 1,, v k are vectors of a vector space V, then is a subspace of V. H = Span{ v 1,, v k } a 3b b a Example 3: Let H be the set of all vectors of the form, a b where a, b are arbitrary real numbers. Determine if H is a subspace of R 4.
Week 8: Vector space and subspace 8 Definition (Linear transformation). The map T : V W is called a linear transformation if T ( u + v) = T ( u) + T ( v); T (c v) = ct ( v) Definition (Kernel). The kernel of T is Definition (Range). The range of T is Theorem: If T : V W is a linear transformation, then Ker(T ) is a subspace of V. Range(T ) is a subspace of W. Proof.
Week 8: Vector space and subspace 9 Kernel for linear transformation T : R n R m Let A be the standard matrix of T, then The kernel (a.k.a null space) of A is the set of all solutions of A x = 0. An explicit description of Nul A: Example 4: Find a spanning set for the null space of the matrix 3 6 1 1 7 A = 1 2 2 3 1 2 4 5 8 4
Week 8: Vector space and subspace 10 Range for linear transformation T : R n R m Let A be the standard matrix of T, then The range (a.k.a column space) of A is the set of all linear combinations of the columns of A. The contrast between Null A and Col A. 2 4 2 1 Example 5: A = 2 5 7 3 3 7 8 6 a. If Col A is a subspace of R k, what is k? b. If Nul A is a subspace of R k, what is k?
Week 8: Vector space and subspace 11 2 4 2 1 Example 6: A = 2 5 7 3 3 7 8 6 a. Find a nonzero vector in Col A. b. Find a nonzero vector in Nul A. Example 7: Let 2 4 2 1 3 A = 2 5 7 3, u = 2 1 3 7 8 6 0 3 and v = 1 3 a. Determine if u is in Nul A. Could u be in Col A? b. Determine if v is in Col A. Could v be in Nul A?
Week 8: Vector space and subspace 12 Contrast between Nul A and Col A for an m n matrix A. Nul A Col A 1. Nul A is a subspace of R n 1. Col A is a subspace of R m 2. Nul A is implicitly defined; 2. Col A is explicitly defined. (A x = 0) 3. Any vector v in Nul A has 3. Any vector v in Col A has the property that A v = 0. the property that A x = v is consistent. 4. Given a specific vector v, 4. Given a specific vector v, it is easy to tell if v is it may take time to tell if v is in Nul A. in Col A. 5. Nul A = { 0} if and only if 5. Col A = R m if and only if A x = 0 has only the trivial A x = b has a solution for every b solution. every b in R m.