6.1 Inner Product, Length & Orthogonality Math 2331 { Linear Algebra 6.1 Inner Product, Length & Orthogonality Jiwen He Department of Mathematics, University of Houston [email protected] math.uh.edu/∼jiwenhe/math2331 Jiwen He, University of Houston Math 2331, Linear Algebra 1 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces 6.1 Inner Product, Length & Orthogonality Inner Product: Examples, Definition, Properties Length of a Vector: Examples, Definition, Properties Orthogonal Orthogonal Vectors The Pythagorean Theorem Orthogonal Complements Row, Null and Columns Spaces Jiwen He, University of Houston Math 2331, Linear Algebra 2 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces Motivation: Example Not all linear systems have solutions. Example 1 2 x 3 No solution to 1 = exists. Why? 2 4 x2 2 Ax is a point on the line spanned by 1 and b is not on the line. So 2 Ax 6= b for all x. Jiwen He, University of Houston Math 2331, Linear Algebra 3 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces Motivation: Example (cont.) Instead find bx so that Abx lies "closest" to b. Using information we will learn in this chapter, we will find that 1:4 1: 4 x= ; so that Ax = . b 0 b 2: 8 Segment joining Abx and b is perpendicular ( or orthogonal) to the set of solutions to Ax = b. Need to develop fundamental ideas of length, orthogonality and orthogonal projections. Jiwen He, University of Houston Math 2331, Linear Algebra 4 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces Inner Product Inner Product Inner product or dot product of 2 3 2 3 u1 v1 6 u2 7 6 v2 7 u = 6 7 and v = 6 7 : 6 . 7 6 . 7 4 . 5 4 . 5 un vn 2 3 v1 6 v2 7 u · v = uT v = u u ··· u 6 7 1 2 n 6 . 7 4 . 5 vn = u1v1 + u2v2 + ··· + unvn v · u = vT u = uT v = u · v Jiwen He, University of Houston Math 2331, Linear Algebra 5 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces Inner Product: Properties Theorem (1) Let u; v and w be vectors in Rn, and let c be any scalar. Then a. u · v = v · u b.( u + v) · w = u · w + v · w c.( cu) · v =c (u · v) = u · (cv) d. u · u ≥ 0, and u · u = 0 if and only if u = 0. Combining parts b and c, one can show (c1u1 + ··· + cpup) · w =c1 (u1 · w) + ··· + cp (up · w) Jiwen He, University of Houston Math 2331, Linear Algebra 6 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces Length of a Vector Length of a Vector 2 3 v1 6 v2 7 For v = 6 7, the length or norm of v is the nonnegative 6 . 7 4 . 5 vn scalar kvk defined by p q 2 2 2 2 kvk = v · v = v1 + v2 + ··· + vn and kvk = v · v: For any scalar c, kcvk = jcj kvk Jiwen He, University of Houston Math 2331, Linear Algebra 7 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces Length of a Vector: Example Example a p If v = , then kvk = a2 + b2 (distance between 0 and v). b Picture: Jiwen He, University of Houston Math 2331, Linear Algebra 8 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces Distance Distance The distance between u and v in Rn: dist(u; v) = ku − vk. Example This agrees with the usual formulas for R2 and R3. Let u = (u1; u2) and v = (v1; v2). Then u − v = (u1 − v1; u2 − v2) and dist(u; v) = ku − vk = k(u1 − v1; u2 − v2)k q 2 2 = (u1 − v1) + (u2 − v2) Jiwen He, University of Houston Math 2331, Linear Algebra 9 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces Orthogonal Vectors [dist (u; v)]2 = ku − vk2 = (u − v) · (u − v) = (u) · (u − v) + (−v) · (u − v) = = u · u − u · v + −v · u + v · v = kuk2 + kvk2 − 2u · v ) [dist (u; v)]2 = kuk2 + kvk2 − 2u · v ) [dist (u; −v)]2 = kuk2 +kvk2 +2u·v Since [dist (u; −v)]2 = [dist (u; v)]2, u · v = . Orthogonal Two vectors u and v are said to be orthogonal (to each other) if u · v = 0. Jiwen He, University of Houston Math 2331, Linear Algebra 10 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces The Pythagorean Theorem Theorem (2 The Pythagorean Theorem) Two vectors u and v are orthogonal if and only if ku + vk2 = kuk2 + kvk2 Jiwen He, University of Houston Math 2331, Linear Algebra 11 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces Orthogonal Complements Orthogonal Complements If a vector z is orthogonal to every vector in a subspace W of Rn, then z is said to be orthogonal to W . The set of vectors z that are orthogonal to W is called the orthogonal complement of W and is denoted by W ? (read as \W perp"). Jiwen He, University of Houston Math 2331, Linear Algebra 12 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces Row, Null and Columns Spaces Theorem (3) Let A be an m × n matrix. Then the orthogonal complement of the row space of A is the nullspace of A, and the orthogonal complement of the column space of A is the nullspace of AT : (Row A)? =Nul A, (Col A)? =Nul AT . Why? (See complete proof in the text) Note that 2 3 2 3 r1 · x 0 6 r2 · x 7 6 0 7 Ax = 6 7 = 6 7 6 . 7 6 . 7 4 . 5 4 . 5 rm · x 0 and so x is orthogonal to the row A since x is orthogonal to r1::::; rm. Jiwen He, University of Houston Math 2331, Linear Algebra 13 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces Row, Null and Columns Spaces: Example Example 1 0 −1 Let A = . 2 0 2 82 3 2 39 < 0 1 = Basis for Nul A = 4 1 5 ; 4 0 5 and Nul A is a plane in R3. : 0 1 ; 82 39 < 1 = Basis for Row A = 4 0 5 and Row A is a line in R3. : −1 ; 1 Basis for Col A = and Col A is a line in R2. 2 −2 Basis for Nul AT = and Nul AT is a line in R2. 1 Jiwen He, University of Houston Math 2331, Linear Algebra 14 / 15 6.1 Inner Product, Length & Orthogonality Inner Product Length Orthogonal Null and Columns Spaces Row, Null and Columns Spaces: Example (cont.) Subspaces Nul A and Row A Subspaces Nul AT and Col A Jiwen He, University of Houston Math 2331, Linear Algebra 15 / 15.