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## 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 = \{(a_{ij}) \in V : a_{ij} = 0,\forall i > j\}##

##V_3 = \{(a_{ij}) \in V: a_{ij} = 0, \forall i \neq j\}##

Prove that for both statements it is true that ##V = V_1 \oplus V_2 \oplus V_3##

Find the projections associated to both decompositions.

## Homework Equations

Projections are linear transformations where ##P^2 = P##.

## The Attempt at a Solution

So for both it was fairly trivial to prove the direct sum. To be explicit about my method, in the first case I took the basis of ##V_1## to be ##\{(1,0,0,0),(0,0,0,1)\}## (from ##\{(1,1,1,0),(0,0,0,1)\}## given by the definition of the subspace) and for ##V_2, V_3## the given vectors. I used these as columns, performed matrix operations to reach row echelon form, and thereby discovered that I could form a diagonal matrix and that each column was linearly independent.

From there, the sum clearly gave me a general vector in ##V##, and by the definition of their linear independence the intersection between the spaces is necessarily ##\{0\}##. Therefore we have a direct sum in the first case.

In the second case, the basis of ##V_1## is a matrix with entries of ##1## above the diagonal and ##0## along and below the diagonal, ##V_2## is ##0## along and above the diagonal and ##1## below, and the basis of ##V_3## is a diagonal matrix with ##0## above and below the diagonal.

Again the direct sum is clear.

**Where assistance is needed**

However, I do not know how to find the projections associated to these decompositions. I do know the definition of a a projection. Is there some intuitive way to see what it would be like here or in similar cases, or do I need to apply the formula ##P_x = A(A^TA)^{-1}A^T##? If so, could I see an example from the first case?