Matematik och Datavetenskap, Chalmers Tekniska Högskola och Göteborgs Universitet

ALA-B, 2001, studio 5.3

Home, w 1, w 2, w 3, w 4, w 5, w 6, w 7. Matlab: analysis, linear algebra.

Linear algebra in R^n, part 2.

Obligatory work: There not enough exercises in the book. Therefore you must write them yourself in such a way that you can read them and repeat them several times. This is a good way to learn. If this does not convince you, then hear this: this stuff will surely be on the exam!

1. Let the column vectors a,b,c in R^4 and the matrix A be as in the previous exercise:

a=[1; 2; 3; 4], b=[4; 3; 2; 1], c=2*a-3*b, A=[a b c]

Quickly repeat problems 4-6 of the previous exercise. We saw there that a,b,c are not linearly independent.

Show that the following are linearly independent:

(a) a,b    hint: B=[a b], solve Bx=0 by Gauss elimination
(b) a,c    hint: C=[a c], solve Cx=0 by Gauss elimination
(c) b,c    hint: D=[b c], solve Dx=0 by Gauss elimination

Since you have already done Gauss elimination by hand several times, you can now save time by using the matlab function rref.

2. Basis, Ch 41.6-9. This means that we have found three bases for the space V=span(a,b,c), namely, a,b and a,c and b,c. Remember that the vector v=[1;1;1;1] belongs to V (studio 5.2, exercise 4). Write v uniquely as a linear combination of

(a) a,b    hint: solve Bx=v (use rref([B v])), then v=x1a+x2b
(b) a,c    hint: solve Cy=v, then v=y1a+y2c
(c) b,c    hint: solve Dz=v, then v=z1b+z2c

3. Linear function, Ch 41.15-19. Let f(x)=Ax with A as before. Show that

(a) f : R^3 -> R^4
(b) f is linear

4. Range R(A), Ch 41.10. The range (värderummet, värdemängden) of the linear function f(x)=Ax is

R(A)={y in R^4 : y=Ax, x in R^3}.

(a) Show that R(A) = V = span(a,b,c).
(b) Show that R(A) is a linear subspace of R^4.
(c) Find more than one basis for R(A). (See exercise 2.)

Answer:  (a)  y = Ax = x1a+x2b+x3c .  (b)  y,v in R(A),  a,b numbers   =>  y=Ax, v=Au   =>  ay+bv =A(ax+bu)   =>  ay+bv in R(A).

5. Null space, Ch 41.11. The null space (nollrummet) of the linear function f(x)=Ax is

N(A)={x in R^3 : Ax=0}.

(a) Show that N(A) is a linear subspace of R^3.
(b) Find a basis for N(A).

Answer:  (a) Solve Ax=0 by Gauss elimination. The solution is x=t[-2;3;1], where t is an arbitrary number. So N(A)=span(g) where g=[-2;3;1] and g is a basis for N(A), i.e., N(A)=span(g).

6. Solution of Ax=y, Ch 41.12. Solve Ax=y with

(a) y=[1;1;1;1]
(b) y=[1;1;1;0]

Answer: (a) x=[0.2;0.2;0]+t[-2;3;1]=xp+xh, where Axp=y and Axh=0, i.e., xh in N(A). Non-unique solution. This means that N(A) is not 0. (b) No solution. This means that y does not belong to R(A).

7. Let the matrices B, C, D be as before. We know that R(B)=R(C)=R(D)=R(A)=span(a,b,c)=V.

(a) Find N(B), N(C), N(D).
(b) Solve Bx=y with y=[1;1;1;1].
(c) Solve Cx=y with y=[1;1;1;1].
(d) Solve Dx=y with y=[1;1;1;1].

Answer: (a) N(B)=N(C)=N(D)=0 (this means the zero space consisting of only the zero vector). (b) x=[0.2;0.2], unique solution because N(B)=0.

8. Transpose of the linear function f(x)=Ax, Ch 41.19.

(a) Show that (AB)T=BTAT. Hint: use formula (41.37).
(b) Show that (Ax,y)= (x,ATy). Hint: remember that (x,y)=yTx and use (a).
(c) Illustrate this by computing (y'*A)*x and (A'*y)'*x for some A,x,y.

/stig



Last modified: Tue Nov 21 15:17:02 MET 2000