高等代数与解析几何英文习题

《高等代数与解析几何》英文习题

主讲老师：林 磊

1. (Feb. 28)

Is ⎭⎬⎫⎩⎨⎧⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝

⎛0101,0110,1112,1011 a basis for the linear space of all 22⨯matrices?

2. (Mar. 1)

Let k j i u 32++=. Find vectors v and w that are both

orthogonal to u and to each other.

3. (Mar. 4)

Let },...,,{21n v v v S = be a basis for a linear space V and let U be

a subspace of V . Is it necessarily true that a basis for U is a subset of S ? Why?

4. (Mar. 7)

In (1)-(2) determine which of the given functions are inner products on 3R where

⎪⎪⎪⎭⎫ ⎝⎛=321u u u α and ⎪⎪⎪⎭

⎫ ⎝⎛=321v v v β

(1) 332211432),(v u v u v u ++=βα;

(2) 132231),(v u v u v u ++=βα.

5. (Mar. 8)

In Exercises (1)-(2) determine whether the given set of vectors is orthogonal, orthonormal, or neither with respect to the Euclidean inner product.

(1) {})3,0(),2,1(;

(2) {})1,0,1(),0,1,0(),1,0,1(-.

6. (Mar. 11)

Compute the area of the triangle with vertices )7,2,0(, )3,5,2(-, and )1,1,1(.

7. (Mar. 14)

Show that ),(4||||22βαβαβα=--+.

8. (Mar. 15)

In Exercises (1) and (2) find an equation for the plane that passes through the point P and that is parallel to the plane whose general equation is given.

(1) )5,3,2(-=P ; 01273=++-z y x .

(2) )1,4,6(-=P ; 06352=+++-z y x .

9. (Mar. 18)

Let T : 32R R → be a linear transformation such that

.15211,34011⎥⎥⎥⎦

⎤⎢⎢⎢⎣⎡=⎪⎪⎭⎫ ⎝⎛⎥⎦⎤⎢⎣⎡-⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎪⎪⎭⎫ ⎝⎛⎥⎦⎤⎢⎣⎡T T (a) Find ⎪⎪⎭⎫

⎝⎛⎥⎦⎤⎢⎣⎡67T ;

(b) Find ⎪⎪⎭⎫

⎝⎛⎥⎦⎤⎢⎣⎡y x T ; (c) Find a matrix A such that

⎥⎦

⎤⎢⎣⎡=

⎪⎪⎭⎫ ⎝⎛⎥⎦⎤⎢⎣⎡y x A y x T . 10. (Mar. 21)

If },,,{21n v v v spans a linear space V , is it possible for },,,{32n v v v 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) 04532=+++z y x ; 02652=-+-z y x .

(2) 02453=-+-z y x ; 01274=+++z y x .

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) 014824222=--++-y x z y x .

(2) 048161863222=-----+-z y x z y x

13. (Mar. 28)

In Exercises (1)-(2) find the matrix representation of the given linear transformation T : 33][x R R →with respect to the ordered bases

{})1,0,0(),0,1,0(),0,0,1(=B for 3R and },,1{'2x x B =for 3][x R .

(1) c bx ax c b a T ++=2)),,((.

(2) c x b a c b a T -+=)()),,((.

14. (Mar. 29)

In Exercises (1)-(2) find bases for the kernel and range of the given linear transformation T : 33][][x x R R →.

(1) b ax c bx ax T +=++2)(2.

(2) x c b c bx ax T )()(2+=++.