Short answer questions No explanation of answers needed for

Short answer questions. No explanation of answers needed for this problem only. Be sure to explain your answers and show your work on all other problems! Give an example of a homogeneous linear system. Give an example of an inconsistent linear system. Give an example of a 4-element set of distinct vectors {v_1, v_2, v_3, v_4} in R^3 that does not span R^3. Give an example of a 3-element set of distinct vectors {v_1, v_2, v_3} in R^3 that is linearly dependent. Give an example of a linear transformation T: R^2 rightarrow R^2. How many solutions can a system of 2013 linear equations in 2014 variables have? TRUE or FALSE: if U_1 is not in the span of {u_2, u_3}, then {u_1, u_2, u_3} is linearly independent.

Solution

(a) here i can give the example for homogeneous linear system

x+y+z=0 (1) eq

x-y+2z=0 (2)eq

3x-y+5z=0 (3)eq

now i am sloving eq (1) and (2)

x+y+z=0

x-y+2z=0

by cancling +y,-y we get 2x+3z=0

2x=-3z

now x==3/2z we got the x value.

now we substitute the x vale in either in eq (1) or (2)

i sub in eq eq(1), x+y+z=0

-3/2z+y+z=0

we get y= 1/2

then sub values x,y in any equation we get z value.

(b) example for inconsistancy linear system

x+y=3

4x+4y=10

theses are examples for incosistancy linear syatem because, it contains no solution.the incosistancy can be seen by multiplying first equation by 4 and subtract second to obtain 0=2

(c) The set spans R^3 if every vector in R^3 can be written as a linear combination of the vectors in the set. The typical way of showing that a set of vectors spans the space they\'re in is to write out a matrix with columns being these vectors:

(1,1,2,) (2,2,0) (2,4,3) (-1,2,6) these are the 4 elements of span of vector.

(d) 3 element of lenear dependency vectors

here are the examples v1=(1,2,-1) v2=(2,-1,1) v3=(8,1,1)

(e ) here is the example of transformation of linear system r^2->r^1

2v1+ v2=4

2v2-3v3=1

by solving this we have solution

(f) i think there many solutions can occur in variables

a11x1+a12x2+dots+a1nxn=b1

a21x1+a22x2+dots+a2nxn=b2

ar1x1+ar2x2+dots+arnxn=br like this many solution may occur.

(g)false.

because u1,u2,u3 are linearly dependant