In regards to your edit, I don't know how to calculate such a matrix. Could you explain to me?
Regarding my answer to the second part, I just took a basis of 1 entries for each subspace (ie strictly upper triang, lower triang, diagonal) and said the projection for each subspace maps that basis...
Hi, StoneTemplePython. Thanks so much for your help. (I think) I eventually figured these problems out with a little help.
In your post you bring artillery into play that I don't quite have yet. In my course, we've covered projections as a build-up to diagonalization and prior to inner product...
Homework Statement
Let ##V = \mathbb{R}^4##. Consider the following subspaces:
##V_1 = \{(x,y,z,t)\ : x = y = z\}, V_2=[(2,1,1,1)], V_3 =[(2,2,1,1)]##
And let ##V = M_n(\mathbb{k})##. Consider the following subspaces:
##V_1 = \{(a_{ij}) \in V : a_{ij} = 0,\forall i < j\}##
##V_2 =...
OK, so I was puzzling over the proof a bit and I think I may have reached something.
Take ##\{v_1 ... v_n\}## to be a basis for ##V## such that ##\{v_1...v_k\}## and ##\{v_{k+1} ... v_n\}## are bases for ##U## and ##W## respectively. Also, given the direct sum, we know the intersection of the...
Homework Statement
[/B]
Let ##V## be a vector space, and let ##U, W## be subspaces of ##V## such that ##V = U \oplus W##. Let ##P_U## be the projection on ##U## in the direction of ##W## and ##P_W## the projection on ##W## in the direction of ##U##. Prove:
##P_U + P_W = Id##, ##P_U P_W = P_W...
Sorry, that was just a typo. It's fixed now, and I find that R4 - R2 gives me a row of 0s and that the rank of that matrix is 3. Is it irrelevant that the linearly dependent vector corresponds to the second column of the matrix, in this case?
Homework Statement
In the follow cases find a maximal linearly independent subset of set ##A##:
(a) ##A = \{(1,0,1,0),(1,1,1,1),(0,1,0,1),(2,0,-1,)\} \in \mathbb{R}^4##
(b) ##A = \{x^2, x^2-x+1, 2x-2, 3\} \in \mathbb{k}[x]##
The Attempt at a Solution
The first part of the exercise is...
I'm sorry, among other things I tried was to show that if ##T^{-1}## is a transformation then it would be linear. But again, I treated it as a function there.
I understand how to show ##T^{-1}## is a subspace (although I'll need to ponder for a moment how to write that the 0 vector belongs...
OK, I've approached the problem differently, though I feel a little stuck.
##B \subset Im(T)## is a subspace.
##Im(T) = \{w \in W : w = T(v)\}##
##B \subset Im(T) = \{b \in W : b = T(a), a \in V\}##
##Ker(T) = \{v \in V : T(v) = 0\}##
##A = T^{-1}(B) \rightarrow A = \{a \in V : T(a) = b\}##...
Sorry, the notation was bugging out. That doesn't read "Ciff" but instead "C iff" and N(T) was an oversight on my part, as I was translating from the Spanish Núcleo de T, which is the Kernel and might just be a Spanish adaptation of the notation denoting the null space? I edited to make the...
Homework Statement
Let ##V## and ##W## be vector spaces, ##T : V \rightarrow W## a linear transformation and ##B \subset Im(T)## a subspace.
(a) Prove that ##A = T^{-1}(B)## is the only subspace of ##V## such that ##Ker(T) \subseteq A## and ##T(A) = B##
(b) Let ##C \subseteq V## be a...
I'll reply to this post later today once I arrive home with my corrected proof that ##\varphi## is surjective, to make sure that my notation is what you were trying to describe.
As far as this unknown bit of notation, the exercise states that ##\mathbb{k}^{\{x\}} \cong \mathbb{k}##. Would that...
Would the demonstration be something like...
Suppose ##\varphi## is injective. In this case if ##\varphi(a_1, a_2, \ldots , a_{ii}) = 0_{\mathbb{k}^n}##, as ##0_{\mathbb{k}^n} = \varphi(0_{D_n(\mathbb{k})})##, we can establish the equality ##\varphi(a_1, a_2, \ldots , a_{ii}) =...
Thanks to both of you! So I've put together this for the first exercise then.
1. ##\varphi : D_n(\mathbb{k}) \rightarrow \mathbb{k}^n, \varphi (a_{11}, a_{22}, \ldots , a_{ii}) = (v_1, v_2, \ldots , v_i)##
2.
##\varphi (a_{11} + b_{11}) = \varphi (a_{11}) + \varphi (b_{11}) = (v_1) + (w_1) =...