- #1
Eclair_de_XII
- 1,083
- 91
Homework Statement
"Show that every subspace of ##ℝ^n## is the set of solutions to a homogeneous system of linear equations. (Hint: If a subspace ##W## consists of only the zero vector or is all of ##ℝ^n##, ##W## is the set of solutions to ##IX=0## or ##0_vX=0##, respectively.
Assume ##W## is not one of these two subspaces. Let ##β=\{v_1,...,v_k\}## for ##W##. Extend this basis to ##β`=β∪\{v_{k+1},...,v_n\}## to span all of ##ℝ^n##. Let ##T: ℝ^n → ℝ^n## be the linear transformation so that ##T(v_1)=T(v_2)=...=T(v_k)=0## and ##T(v_j)=I(v_j)## for ##k+1≤j≤n##, where ##I(v)## is the identity function. Now use ##T## to obtain a matrix A so that ##W## is the set of solutions to the homogeneous system ##AX=0_v##.)"
Homework Equations
##T_\alpha=PT_βP^{-1}##
where ##P## is the transition matrix from basis ##β## to basis ##α##.
The Attempt at a Solution
##T(c_1v_1+...+c_kv_k)=T(c_1v_1)+...+T(c_kv_k)=c_1T(v_1)+...+c_kT(v_k)=0_v##
##T(c_{k+1}v_{k+1}+...+c_nv_n)=c_{k+1}T(v_{k+1})+...+c_nT(v_n)=c_{k+1}v_{k+1}+...+c_nv_n=0_v##
I honestly don't know what I'm doing here, and if anyone would like to provide feedback on what I'm doing wrong, or what I should actually be doing instead of this, that would be much appreciated. Do I have to left-multiply the ##n×n## transformation matrix by a column vector whose entries consist of the constants ##c_i##, proving that the product is the zero vector, and that it solves the homogeneous system? ##\left[T(v)\right]\left[c_i\right]=\left[0_v\right]##. Something like that, maybe? Sorry, and thanks.
Last edited: