Inner product • Inner product in R2. Angle between vectors. • Examples of inner product spaces. • Orthogonal vectors. • Orthogonal subspaces • Example of the four fundamental subspaces. • Orthogonal projections and orthonormal vectors 2 In R the inner product of x = (x1, x2), y = (y1, y2) is hx, yi = x1y1 + x2y2. Inner product is symmetric: hx, yi = hy, xi Question: Is inner product invariant under rotation? Proof: Consider rotation matrices Hence without loss of generality we can consider inner product of vectors x = (x1, x2), y = (y1, 0) Check:If x = y, then hx, yi = hx, xi = |x|2. Check Inner product of x, y is a product of length of y and projection of x on y Inner product of x, y is a product of length of x and projection of y on x. Why? 2 2 Projection on a vector a = (a1a2), a1 + a2 = 1 2 a1 a1a2 x1 Ax = ha, xia 2 = a1a2 a2 x2 Definition For a vector space V an inner product is an additional function h, i : V × V → R( or C) Remark: We concentrate now on real case, a completely analogous definition for complex case is put in brackets. h, i satisfies: 1. linearity hz, αx + βyi = αhz, xi + βhz, yi 2. Symmetry: hx, yi = hy, xi (conjugate symmetry for complex case hx, yi = hy, xi) 3. Positive definiteness: hx, xi ≥ 0, hx, xi = 0 if and only if x = 0 Question: Why property 3. makes sense in complex case? Example 1: Define an inner (scalar) product in Rn t X hx, yi = x y = xiyi. 1 Example 2: For a matrix A : Rn → Rn define a “inner product” of x y as hx, yiA = hx, Ayi Which matrices A can we pick so that hx, yiA is a true inner product? Example 3: For functions f(x) g(x) on [0, 1] define Z 1 hf, giL2 = f(x)g(x)dx. 0 Question: Which functions on [0, 1] belong to the above-determined (linear) inner product subspace? Note: Inner product is a new additional algebraic structure on a vector space. A vector space with an inner product is called an inner product space. Question: For a given inner product can define length of a vector? angle between two vectors? projection of one vector on another? Cauchy-Schwarz inequality: hx, yi ≤ |x||y|. Proof: The length of projection of x on y never exceeds |x|. Definition: Vectors x, y are orthogonal if hx, yi = 0 Reinterpret this definition using projections Question: Give an example of orthogonal functions on [0, 1] Definition: Two vector subspaces W ,U of a vector space V , i.e. W ⊂ V , U ⊂ V are othogonal if for any x ∈ U and y ∈ W hx, yi = 0. We denote it as U ⊥ W Question: Which vectors are contained both in W and U, if U ⊥ W ? Question: Give an example (if possible) of two orthogonal planes (two-dimensional subspaces) in R3 Fundamental theorem of Algebra Part II: In the real case R(At) ⊥ N (A), R(A) ⊥ N (At) Corollary: Every matrix transforms its row space into its column space. Question: How to determine a projection of a vector x on a vector subspace W ? Hint: start with case 1, suppose W =span{x}. Hint for general case is the following definition. Definition A set of vectors x1, x2, . , xk is said to be orthonormal if ( 1 if i = j, hxi, xji = δij = 0 if i 6= j. 2.