E231 HW 3, due Friday, September 30, 5:00 PM1. Problem 9 from HW 2.2. Problem 11 from HW 2.3. Let V and W be finite dimensional vector spaces over the same field, and letA ∈ L (V, W ). Verify the following(a) If {vi}ni=1is a basis for V , thenspan {A(v1), . . . , A(vn)} = Range(A)(b) dim [Range(A)] ≤ dimV(c) If A is onto, then dim(W ) ≤ dim(V ).(d) If dim(W ) < dim(V ), then A is not 1-to-1.(e) If A is 1-to-1, then dim(W ) ≥ dim(V ).(f) If A is 1-to-1 and onto, then dim(W ) = dim(V ).(g) Suppose dim(W ) = dim(V ). Then A is 1-to-1 if and only if A is onto.4. Suppose that A ∈ Cm×n. View A as a linear operator from Cn→ Cmusingstandard matrix-vector multiplication. Show that the induced 1-norm of A is themaximum column sumkAk1,1= max1≤j≤nmXi=1|Aij|In your derivation, show how to construct a vector x ∈ Cnsuch thatkxk1= 1, kAxk1= kAk1,1Using Matlab, check the syntax of norm, and describe how to use Matlab to com-pute this norm.5. Suppose that A ∈ Cm×n. View A as a linear operator from Cn→ Cmusingstandard matrix-vector multiplication. Show that the induced ∞-norm of A is themaximum row sumkAk∞,∞= max1≤i≤mnXj=1|Aij|In your derivation, show how to construct a vector x ∈ Cnsuch thatkxk∞= 1, kAxk∞= kAk∞,∞Using Matlab, check the syntax of norm, and describe how to use Matlab to com-pute this norm.16. Suppose that x and y are elements of a normed vector space. Show that| kxk − kyk | ≤ kx − yk7. Consider the vector space R2over the field R. Recall that for a norm k·k, the unitball is{v ∈ V : kvk ≤ 1}(a) Draw the unit ball using the k·k1norm in R2.(b) Draw the unit ball using the k·k2norm in R2.(c) Draw the unit ball using the k·k4norm in R2.(d) Draw the unit ball using the k·k8norm in R2.(e) Draw the unit ball using the k·k∞norm in R2.8. Let V be a finite dimensional inner product space. Suppose that E and F aresubspaces of V .(a) Show that E⊥is a subspace of V(b) Show that E ⊂ F if and only if F⊥⊂ E⊥.9. V is a finite dimensional inner product space over C. Suppose {vikni=1is an or-thonormal basis set for V . Also suppose {wikni=1is an orthonormal basis set forV . Since both sets are basis sets, they each span V . Hence for each i, there existcomplex numbers qki, k = 1, . . . , n, such thatvi=nXk=1qkiwkLet Q be the n × n complex matrix whose (i, j) entry is qij. Show that Q∗Q = In.10. Let V be Cn[a, b] (i.e., the set of n × 1 vectors of complex-valued, continuousfunctions defined on the real interval [a b]). For f, g ∈ V , the inner product ishf, gi :=Zbaf∗(t)g(t)dtLet W be Cm, the set of m × 1 column vectors of complex numbers. For x, y ∈ W ,the inner product ishx, yi := x∗ySuppose Φ : [a b] → Cm×nbe a continuous function (Φ is a m × n matrix ofcontinuous functions, each defined on [a b]). Define a linear map A : V → W by:if f ∈ V , then A(f) isA(f) :=ZbaΦ(t)f(t)dt(a) Find the adjoint A∗.(b) Find A ◦