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1、高等代数与解析几何英文习题主讲老师：林磊1. (Feb. 28) a basis for the linear space ofall 22matrices?Let w = Z + 2 J + 3 .2. (Mar. 1)Find vectors v and w that are bothorthogonal to u and to each other*.3. (Mar. 4)Let S = vi,v2,.,vti be a basis for a linear space V and let U be a subspace of V. Is it necessarily true that
2、 a basis for UiSa subset of S? Why?4. (Mar. 7)In (1)-(2) determine which of the given functions are inner products on R3 where/%a = u2件3a、and = v1儿(1) (a,y?) = 2w1vl + 3m2v2 4w3v3;(2) (a,) = w1v3 + w2v2 + w3v1.5. (Mar. 8)In Exercises (1)-(2) determine whether the given set of vectors is orthogonal,
3、orthonormal, or neither with respect to the Euclidean inner product.(1) d,2), (0,3)；(2)(i,oa(0,1,0), (-,o,i).6. (Mar. 11)Compute the area of the triangle with vertices (0,2,7), (2,-5,3), and (1,1,1).7. (Mar. 14)Show that a + 2 一Ia一万2=4()8. (Mar. 15)In Exercises (1) and (2) find an equation for the p
4、lane that passes through the point P and that is parallel to the plane whose general equation is given.(1) P = (2,3-5)； 3x-7y + 2z + l = 0.(2) P = (-6,4,l); -2x + 5y + 3z + 6 = 0.9. (Mar. 18)Let T: R2 R3 be a linear transformation such that(a) Find T(b) Find T ;Lyy(c) Find a matrix A such thatWH10.
5、(Mar. 21)If vpv2,.,vwspans a linear space V, is it possible for v2,v3,.,vzj to span V ? Explain your answer.11. (Mar. 22)In Exercises (1) and (2) find parametric equations for the line of intersection between the planes whose general equations are given.(1) 2x + 3y+ 5z + 4 = 0； 2x-5y + 6z-2 = 0.(2)
6、3x-5y+ 4z-2 = 0; 4x + 7y+ 2z +1 = 0.12. (Mar. 25)In Exercises (1)-(2) name the surface determined by the given equation and give its equation in a coordinate system in which the surface is in standard position.(1) 4x2 - y2 + 2z2 + 8x - 4y -1 = 0.(2) -x2 +3y2 -z2 -6x-18y-16z-48 =013. (Mar. 28)In Exer
7、cises (1)-(2) find the matrix representation of the given linear transformation T: R3 R3 with respect to the ordered basesB = (l,0,0), (0,1,0), (0,0,1) for R3 and B = l,x,x2for R3.(1) T(4,O, C) = Or2+7 + c.(2) T(a,b,c) = (a-b)x-c.14. (Mar. 29)In Exercises (1)-(2) find bases for the kernel and range of the given linear transformation T: R3 R3.(1) T(ax2 +bx + c) = 2ax + b.(2) T(ax2 +bx + c) = S + c)x.
