Prove properties 2, 3 and 4 of Theorem 7.(Reference

Problem 62E Determine n and m so that In A = A and A Im = A, where: (4 × 2)

Problem 2E The matrices and vectors in Eq Perf to venry the given equality or none quality. (FE)D = F(ED) eq

Problem 1E The matrices and vectors in Eq Perf to venry the given equality or nonequality. (DE)F = D(EF) eq

Problem 3E The matrices and vectors in Eq Perf to venry the given equality or none quality. DE ? ED eq

Problem 4E The matrices and vectors in Eq Perf to venry the given equality or nonequality. EF ? FE eq

Problem 5E The matrices and vectors in Eq Perf to venry the given equality or nonequality. Fu = Fv eq

Problem 6E The matrices and vectors in Eq Perf to venry the given equality or nonequality. 3F = 7Fv eq

Problem 7E Find The matrices AT eq

Problem 12E Find The matrices (EF)v eq

Problem 11E Find The matrices (Fv)T eq

Problem 8E Find The matrices. DT eq

Problem 10E Find The matrices AT C eq

Problem 9E Find The matrices ET F eq

Problem 13E Calculate the scalars. uTv eq

Problem 14E Calculate the scalars. vT Fu eq

Problem 15E Calculate the scalars. vT Dv eq

Problem 16E Calculate the scalars. vT Fv eq

Problem 17E Calculate the scalars. uTu eq

Problem 18E Calculate the scalars. vTv eq

Problem 19E Calculate the scalars. ||u|| eq

Problem 20E Calculate the scalars. ||Dv|| eq

Problem 22E Calculate the scalars.||u – v|| eq

Problem 23E Calculate the scalars. ||Fu|| eq

Problem 21E Calculate the scalars. ||Au|| eq

Problem 24E Calculate the scalars. ||Fv|| eq

Problem 25E Calculate the scalars. ||(D – E)u|| eq

Problem 26E Let A and B be (2 × 2) matrices. Prove or find a counterexample for this statement: (A- B)(A + B) = A2 – B2

Problem 27E Let A and B be (2 × 2) matrices such that A2 = AB and A ? 0. Can we assert that, by cancellation, A = B? Explain.

Problem 30E Find (2 × 2) matrices A and B such that A and B are symmetric, but AB is not symmetric. [Hint:(AB)T = BF AT = BA]

Problem 28E Let A and B be (2 × 2) matrices such that A2 = AB and A ? 0. Can we assert that, by cancellation, A = B?. Find the flaw in the following proof that A = B. Since A2= AB, A2 – AB = 0, Factoring yields A(A – B) = 0. Since A ? 0, it follows that A – B = 0.Therefore, A = B

Problem 29E Two of the six matrices listed in Eq are symmetric. Identify these matrices, eq

Problem 31E Let A and B be (n × n) symmetric matrices. Give a necessary and sufficient condition for AB to be symmetric. [Hint: (AB)T = BF AT = BA]

Problem 32E Let G be the (2 × 2) matrix that follows, and consider any vector x in R2 where both entries are not simultaneously zero: |x1| + |x2| > 0. Show that xTGx> 0. [Hint: Write xTG xas a sum of squares.]

Problem 33E Find (2 × 2) matrices A and B such that A and B are symmetric, but AB is not symmetric. [Hint:(AB)T = BF AT = BA]

Problem 34E Let A and B be (n × n) symmetric matrices. Give a necessary and sufficient condition for AB to be symmetric. [Hint: (AB)T = BF AT = BA]

Problem 35E Let G be the (2 × 2) matrix that follows, and consider any vector x in R2 where both entries are not simultaneously zero: |x1| + |x2| > 0. Show that xTGx> 0. [Hint: Write xTGxas a sum of squares.].

Problem 36E Using the matrix D in Eq,in Place of G. eq

Problem 37E For F Show that xT Fx ? 0 for all x in R2. Classify those vectors that xT Fx = 0 eq

Problem 40E If x and y are vectors in Rn then the product xTy is often called an inner product. Similarly, the product xyT is often called an outer product. Concem outer products; the matrices and vectors are given in eq v(Ev)T eq

Problem 38E If x and y are vectors in Rn then the product xTy is often called an inner product. Similarly, the product xyT is often called an outer product. Concem outer products; the matrices and vectors are given in eq uvT eq

Problem 41E If x and y are vectors in Rn then the product xTy is often called an inner product. Similarly, the product xyT is often called an outer product. Concem outer products; the matrices and vectors are given in eq u(Ev)T eq

Problem 39E If x and y are vectors in Rn then the product xTy is often called an inner product. Similarly, the product xyT is often called an outer product. Concem outer products; the matrices and vectors are given in eq u(Fu)T eq

Problem 43E If x and y are vectors in Rn then the product xTy is often called an inner product. Similarly, the product xyT is often called an outer product. Concem outer products; the matrices and vectors are given in eq (Av)(Au)T eq

Problem 42E If x and y are vectors in Rn then the product xTy is often called an inner product. Similarly, the product xyT is often called an outer product. Concem outer products; the matrices and vectors are given in eq (Au)(Av)T eq

Problem 45E (Reference Theorem 8)

Problem 46E (Reference Theorem 7)

Problem 44E Let a and b be given by a) If AT + B = C, what is A? b) If AT B = C, what is A? and c) Calculate BC1 B1TC, (BC1)I C2, and ||CB2||

Problem 47E Let A, B,C, and D be matrices such that AB = D and AC - D. The following statements are steps in a proof that if r and s are scalars, then A(rB+sC) = (r + s)D. Use Theorems to provide reasons for each of the steps. a) A( rB + sC ) = A(rB)+ A(sC). ________________ b) Ar(B+ sC) = r(AB) + s(AC) = rD + sD. ________________ c) A(rB + sC) = (r + s)D.

Problem 48E Let x and y be vectors in Rn such that ||x|| = ||y|| = 1 and xTy = 0. Use to show that

Problem 49E (Reference Theorem 10)

Problem 50E Let A be the (2 × 2) matrix Choose some vector b in R2 such that the equation Ax = b is inconsistent. Verify that the associated equation ATAx = AT bis consistent for your choice of b. Let x* be a solution to ATAx= ATb, and select some vectors x at random from R2. Verify that || Ax* - b|| ? || Ax - b || for any of these random choices for x. (In Chapter 3, we will show that AT Ax = AT b is always consistent for any (m × n) matrix A regardless of whether Ax = b is consistent or not. We also show that any solution x* of ATAx = AT b satisfies ||Ax* - b|| ? ||Ax - b|| for all x in Rn that is, such a vector x* minimizes the length of the residual vector r = A

Problem 51E Prove properties 2, 3 and 4 of Theorem 7. (Reference Theorem 7)

Problem 53E Prove properties 2 and 3 of Theorem 8. (Reference Theorem 8)

Problem 52E Let Q be an (m × n) matrix and x any vector in Rn. Prove that xTQTQx? 0. [Hint: Observe that Qxis a vector in Rm.]

Problem 54E Prove properties 2, 3 and 4 of Theorem 9. (Reference Theorem 9)

Problem 56E Prove properties 2, 3, and 4 of Theorem 9.

Problem 57E Prove properties 1 and 3 of Theorem 10.

Problem 55E Prove properties 1 and 3 of Theorem 10. (Reference Theorem 10)

Problem 60E Determine n and m so that In A = A and A Im = A, where: (4 × 4)

Problem 58E Determine n and m so that In A = A and A Im = A, where: A is (2 × 3)

Problem 59E Determine n and m so that In A = A and A Im = A, where: (5 × 7)

Problem 61E Determine n and m so that In A = A and A Im = A, where: A is (4 × 6)