I'm studying to linear algebra in community college, and I'm having a hard time with this exercise because I don't really understand what we are doing.
We have a 3x3 matrix A, and we know that the eigenspaces of that matrix are E(1,A)=Sp(v1,v2) and E(2,A)=Sp(v3) where v1=(1,0,1) v2=(0,1,0) and v3=(1,0,-1).
Then we are supposed to solve an equation of the form Ax=b where b is a vector b=3,4,5.
Am I supposed to create a matrix out of the v1,v2,v3 vectors and then somehow solve x from that? Also i dont know what matrix A is, We just assume to know its eigenspaces (and values), could I solve what matrix A is from the eigenspaces (I dont know how to do that lol).
Any help with this problem would be greatly appreciated, since I don't really know what steps to take and where.
Seems that, from your problem statement:
$$Av_1 = v_1 \,\,\text{ and }\,\, A v_2 = v_2 \,\, \text{ and }\,\, Av_3 = 2v_3.$$
Also, it should be clear that $v_1,v_2,v_3$ span $\mathbb{R}^3$ so
$$b= c_1 v_1 + c_2 v_2 + c_3 v_3$$ for some $c_1,c_2,c_3$ not all zero.
Find the $c_j$'s . Then
$$Ab = A(c_1v_1 + c_2v_2 + c_3 v_3)= c_1 Av_1 + c_2 Av_2 + c_3 Av_3=c_1v_1+c_2v_2+ 2c_3 v_3.$$