Question

For what values of h is the vector u in the span of v and w?

Answer 1

Can't figure out how to make matrices on here so u=(-2,0,h). v=(1,2,3), and w=(0,1,2). As matrices they're obviously all 3x1.

Answer 2

Row reduced it looks like
1 0 -2
0 1 4
0 0 h-2

Answer 3

when does 0 = a number other than 0 ?

Answer 4

Well I know it'd be inconsistent if h isn't 2, but is u in the span of v and w when h=2 or is there no solution?

Answer 5

h is a linear combination of u and v (is in the span of) when the bottom row is all zeros. so h=2 would be good if youve done the rref correctly :)

Answer 6

im of corurse misreading a few things .. names of vectors

Answer 7

Yea the rest was easy enough, but I'm just confused as to what a row of zeroes represents in this situation. It doesn't indicate a free variable because there are only 2 variables each with a solution. Is it just an arbitrary statement that keeps the matrix the same dimensions?

Answer 8

a row of zeros at the bottom represents a set of vectors such that one of them is a linear combination of the others

Answer 9

if we do not get a row of zeros, then that means that the vectors would have been independant of each other

Answer 10

Yea I can see how it all makes sense, I guess I was just confused as to what the zeroes would mean. But that makes sense, I know I read in my book that a row of zeroes always indicated dependence.

Answer 11

thank ya

Answer 12

take 2 vectors from a given plane that are not scalar multiples of each other (they clearly point in different direction).

we can define every other vector in that plane using just scalars of those 2 vectors; any other vector in the plane would ultimately depend on some combination of a and b for their existence

Answer 13

good luck :)

Answer 14

awesome