Ytukyg

Determine whether the sets spans in $R^2$:

a) $S={(1,2),(-1,1)}$

b) $S={(0,0),(1,1)}$

c) $S={(2,4),(-1,2)}$

d) $S={(1,3),(2,-3),(0,2)}$

What I did:

Let be $u=(u_1,u_2)$ any vector en $R^2$ y let be $c_1,c_2,c_3$ scalars then:

a) $u=(u_1,u_2)=c_1(1,2)+c_2(-1,1)=(c_1-c_2,2c_1+c_2)$ which gives the system: $$c_1-c_2=u_1$$
$$2c_1+c_2=u_2$$ The coefficient matrix of the system has determinant 3 so it have a unique solution and therefore, any vector any vector in $R^2$ can be written as a linear combination of vectors of $S$, and therefore, the set $S$ spans in $R^2$.

Am I right?

b) $u=(u_1,u_2)=c_1(0,0)+c_2(1,1)=(c_2,c_2)$ which gives the system: $$c_2=u_1$$
$$c_2=u_2$$ The coefficient matrix of the system has determinant $0$.

In here, the system either has a no solution or infinite solutions, right? And by that can I conclude that the set $S$ doesn't span in $R^2$? Or not? If so, what I'm supposed to do here?

c) $u=(u_1,u_2)=c_1(2,4)+c_2(-1,2)=(2c_1-c_2,4c_1+2c_2)$ which gives the system: $$2c_1-c_2=u_1$$
$$4c_1+2c_2=u_2$$ The coefficient matrix of the system has determinant 8 so it have a unique solution and therefore, any vector any vector in $R^2$ can be written as a linear combination of vectors of $S$, and therefore, the set $S$ spans in $R^2$.

Am I right?

d) $u=(u_1,u_2)=c_1(1,3)+c_2(2,-3)+c_3(0,2)=(c_1+2c_2,3c_1-3c_2+2c_3)$ which gives the system: $$c_1+2c_2=u_1$$
$$3c_1-3c_2+2c_3=u_2$$

I'm stuck in this one. Don't know what I'm supposed to do here?

Please help! Thanks.

You seem to want to prove (a) and (c) twice! First, you give an explicit way of expressing $u$ in terms of the elements of $S$; this proves that $S$ is a spanning set. Then, you compute the determinant of the coefficient matrix; this is non-zero, which also proves that $S$ is a spanning set.

You only need one of these!

In case (b), the determinant is zero, therefore $S$ is not a spanning set.

In case (d), it is enough to pick two elements of $S$ and show that they constitute a spanning set. Then $S$ surely is a spanning set.

Hint: If $V $ is a finite dimensional vector space with $dimV=n$,then any subset of $V $ which contains more than $n$ vectors is linearly dependent. So for the last case,it is enough to check whether any two vectors span $mathbb{R}^2$.Other cases seem right to me.

To determine whether a set spans vector space, all you need to do is to show that every element in that vector space can be written as a linear combination of the elements in the span, this is due to the definition of a spanning set. Furthermore, I will not prove it, but you should try to prove it, it is that if you have n vectors from $F^n$ that span a subspace (or a vector space, since every vector space is its own subspace) then you can put them in a matrix and perform elementary row operations on them, the new set still spans the same space. So, to show a set spans $R^2$, you can show that that spanning set is equal to the spanning set consisting of (1,0) and (0,1), the standard bases. But they don't have to be (1,0) and (0,1).