Linear Algebra Answers

1.find a number T such that (3,1,4),(2,-3,5),(5,9,t) is not linearly independent in R³.

2.let v be the subspace of R⁵ defined by v={(x1,x2,x3,x4, X5)€R⁵:2x1=x2 and X3=X5}

2.1.find a basis of v.

2.2.find a subspace w of R⁵ such that R⁵=v©w.

3.suppose v1,v2,...VM are finite-dimensional subspace of v.prove that v1+v2+...+VM is finite-dimensional and dim(v1+v2+....+VM)is greater or equal dimv1+dimv2+...+dimvm.

1.determine whether the set S is subspace of R⁵ defined by S={(x1,X2,X3,x4,X5)€R⁵:x1=3x2 and X3=7x4}.

2.let S be a subset of f³ defined as S={(x,y,z)€F³:x+y+2z-1=o},then determine S is a subspace of f³ or not.

3.suppose v is a subspace of v.then show that v+v=v.

4.suppose v={(x,y,x+y,x-y,2x)€f⁵:x,y€f}.find a subspace w of f⁵ such that f⁵=v©w.

1.Suppose A and B are both non zero real numbers.find real numbers C and D such that 1/a+ib=c+id.

2.suppose v,w€v.explain why there exists a unique x€v such that v+3x=w.

3.find all values of alpha€c such that alpha(1+I;2-i)=(2+2i;2-i).

4.let -infinity and positive infinity denote two distinct objects,neither of which is in R. Define an addition and scalar multiplication on RU{+infinity} U{-infinity}. specifically,the sum and product of two real numbers is as usual, and for T€R define

T(+infinity)=[-infinity if t<0

[0 if t=0

[+Infinity if t>o

T(-infinity)=[+infinity if t<0

[0 if t=o

[-infinity if t>0,

T+infinity=infinity+t=infinity,t+(-infinity)=-infinity+t=-infinity.

Infinity+infinity=infinity,(-infinity)+(-infinity)=-infinity, infinity+(-infinity)=o

Determine whether RU(infinity)U (-infinity) is a vector space over R.

Let 1 and 1 denote two distinct objects, neither of which is in R.

Define an addition and scalar multiplication on R [ f1g [ f1g as you

could guess from the notation. Specifically, the sum and product of two

real numbers is as usual, and for t 2 R define

t1 D

8

ˆ<

ˆ:

1 if t < 0;

0 if t D 0;

1 if t > 0;

t .1/ D

8

ˆ<

ˆ:

1 if t < 0;

0 if t D 0;

1 if t > 0;

t C1D1C t D 1; t C .1/ D .1/ C t D 1;

1C1D1; .1/ C .1/ D 1; 1 C .1/ D 0:

Is R [ f1g [ f1g a vector space over R? Explain.