Show that a set of vectors spans a subspace

In summary: It starts with an arbitrary vector in the plane and assumes that it can be written as ##\alpha (1,2,3) + \beta (3,4,5)## and then solves for the two unknowns.The first equation is just the vector itself, and the second equation is the vector squared. So, they can solve for the two unknowns by solving these two equations for the two unknowns.
  • #1
mattqchou
2
0

Homework Statement


Show that {(1, 2, 3), (3, 4, 5), (4, 5, 6)} does not span R3. Show that it spans the subspace of R3 consisting of all vectors lying in the plane with the equation x - 2y + z = 0.

Homework Equations

The Attempt at a Solution


I made a matrix of:
A = [ 1 3 4 ; 2 4 5; 3 5 6] and reduced it to row-echelon form to determine rank. I found that rank(A) = 2. Therefore, it can't span R3, since rank(A) < n for Rn.

But then I need to show that it spans the plane given by the equation x - 2y + z = 0.

I thought that perhaps rewriting the problem as: z = -x + 2y and plugging back in for z. But that doesn't really tell me anything.

I also tried to show the span by demonstrating where it has linear combinations by writing span(F) = span{ax - 2by + cz = 0} but I don't know what to do from here.

I was able to find the solution in the solutions manual, but I can't decipher what it's saying or how it did it. It's not using a method described in the book.

lateximg_large.png


Could someone explain this better or give me another method to figure out whether a set spans a subspace?
 

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  • #2
They find explicit coefficients for the vectors (1,2,3) and (3,4,5) to express every vector in the plane as linear combination of these vectors.
The coefficients are (-2x+3y/2) and (x-y/2). Multiply them with the vectors, simplify and you get a general vector in the discussed plane.
 
  • #3
mfb said:
They find explicit coefficients for the vectors (1,2,3) and (3,4,5) to express every vector in the plane as linear combination of these vectors.
The coefficients are (-2x+3y/2) and (x-y/2). Multiply them with the vectors, simplify and you get a general vector in the discussed plane.
Hi,

Could you explain how you find explicit coefficients? I've never heard of this method in my class before. I also can't find that in the book or the glossary.

Thanks!
 
  • #4
See the calculation above, it does exactly this.
It starts with an arbitrary vector in the plane and assumes that it can be written as ##\alpha (1,2,3) + \beta (3,4,5)## and then solves for the two unknowns.
 
  • #5
mattqchou said:
But then I need to show that it spans the plane given by the equation x - 2y + z = 0.

I thought that perhaps rewriting the problem as: z = -x + 2y and plugging back in for z. But that doesn't really tell me anything.
Sure it does. You now have three equations that describe the vectors that lie in that plane:
\begin{align*}
x &= s \\
y &= t \\
z &= -s + 2t
\end{align*} or in vector form,
$$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = s\begin{pmatrix} 1 \\ 0 \\ -1\end{pmatrix} + t\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}.$$ Can you take it from there?
 

Related to Show that a set of vectors spans a subspace

What is a subspace?

A subspace is a subset of a vector space that satisfies all the properties of a vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector.

What does it mean for a set of vectors to span a subspace?

A set of vectors spans a subspace if every vector in that subspace can be written as a linear combination of the given vectors. In other words, the given vectors can be used to create any vector in the subspace.

How do you show that a set of vectors spans a subspace?

To show that a set of vectors spans a subspace, we can use the definition of span and show that every vector in the subspace can be written as a linear combination of the given vectors. This can be done by setting up a system of equations and solving for the coefficients of the linear combination.

What is the difference between a spanning set and a basis?

A spanning set is a set of vectors that can be used to create any vector in a subspace, while a basis is a specific type of spanning set that is also linearly independent. This means that a basis is the smallest possible spanning set for a subspace.

Can a set of vectors span more than one subspace?

Yes, it is possible for a set of vectors to span more than one subspace. This can happen if the given vectors can be used to create linear combinations that result in vectors that are in different subspaces.

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