《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)

0 a basis for the linear space of

all 2 2 matrices? 2. (Mar. 1)

orthogonal to u and to each other. 3. (Mar. 4)

Let S {v 1,v 2,...,v n } be a basis for a linear spaceV and let U be a subs pace of V . Is it n ecessarily true that a basis for U is a subset of S? Why? 4. (Mar. 7)

In (1)-(2) deter mine which of the give n fun cti ons are inner p roducts on R 3 where

u 1

u 2 and

u 3

5. (Mar. 8)

1 1

2 Is

0 1,1

Let u i 2j

3k . Find vectors v and w that are both

V 1 V 2 V 3

(1)(,) 2u 1V 1 3u 2V 2 4u 3V 3;

(2) ( , )

U 1V 3 u 2V 2 u 3V 1 .

In Exercises (1)-(2) determine whether the given set of vectors is orthog on al, orth onor mal, or n either with res pect to the Euclidea n inner p roduct.

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

(2) (1,0,1), (0,1,0), ( 1,0,1).

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 | |2| |2 4(, ).

8. (Mar. 15)

In Exercises (1) and (2) find an equatio n for the plane that p asses through the point P and that is parallel to the plane whose general equati on is give n.

(1) P (2,3, 5)；3x 7y 2z 1 0.

⑵P ( 6,4,1); 2x 5y 3z 6 0 .

9. (Mar. 18)

Let T: R2R3be a lin ear tran sformatio n such that

⑻ Find T

1

1

x

(b) Find T

y

(c) Find a matrix A such that

10. (Mar. 21)

If {v1,v2, ,v n} spans a linear space V , is it possible for {V2,V3, ,V n}

to span V ? Explain your answer.

11. (Mar. 22)

In Exercises (1) and (2) find parametric equations for the line of in tersecti on betwee n the planes whose gen eral equati ons are give n.

(1) 2x 3y 5z 4 0; 2x 5y 6z 2 0.

(2) 3x 5y 4z 2 0; 4x 7y 2z 1 0.

12. (Mar. 25)

In Exercises (1)-(2) name the surface determined by the given equati on and give its equati on in a coord in ate system in which the surface is in sta ndard p ositi on.

(1) 4x2 y2 2z2 8x 4y 1 0.

(2) x2 3y2 z2 6x 18y 16z 48 0

13. (Mar. 28)

In Exercises (1)-(2) find the matrix rep rese ntatio n of the give n linear

o

transformation T : R R[x]3with respect to the ordered bases

B (1,0,0), (0,1,0), (0,0,1) for R3 and B' {1, x,x2} for R[X]3.

(1) T((a,b,c)) ax2 bx c.

⑵ T((a,b,c)) (a b)x c.

14. (Mar. 29)

In Exercises (1)-(2) find bases for the kernel and range of the give n linear transformation T : R[x]3R[x]3.

(1) T(ax2 bx c) 2ax b .

(2) T(ax2 bx c) (b c)x.