Finding the formula of a projection in M2x2 (R)

[benlinus]

Homework Statement

Let T : M_2x2(ℝ) → M_2x2(ℝ) denote the projection on W along U.

W={
[a b]
[c d] : a+b+c+d=0 }

U= span{ I₂ }

Find the matrix representation of T with respect to the standard basis of M_2x2(ℝ) and the formula for T.

Homework Equations

From my notes, I know W is the range of T and U is the kernel of T.

M_2x2(ℝ) = W $\oplus$ U (Direct sum)

The Attempt at a Solution

I am unsure how to proceed. Any thoughts?

 

[mighty2000]

Thank you for your question. I would approach this problem by first understanding the definitions of range and kernel, as well as the concept of a projection.

The range of a linear transformation T is the set of all possible outputs that T can produce. In this case, we are given that W is the range of T. This means that any matrix in W can be obtained as the output of T.

The kernel of T, also known as the null space, is the set of all inputs that produce a zero output. In this case, we are given that U is the kernel of T. This means that any matrix in U will result in a zero output when fed into T.

Next, we can use the formula for a projection to find the matrix representation of T. The formula for a projection is P = A(A^T A)^-1 A^T, where A is the matrix representation of the transformation. In this case, we can let A be the standard basis of M2x2(ℝ), which is given by the matrices [1 0], [0 1], [0 0], [0 0].

We can now plug in the values for A to get the matrix representation of T:

P = [1 0 0 0]([1 0 0 0]^T [1 0 0 0] [1 0 0 0]^T)^-1 [1 0 0 0]^T
[0 1 0 0]([0 1 0 0]^T [0 1 0 0] [0 1 0 0]^T)^-1 [0 1 0 0]^T
[0 0 1 0]([0 0 1 0]^T [0 0 1 0] [0 0 1 0]^T)^-1 [0 0 1 0]^T
[0 0 0 1]([0 0 0 1]^T [0 0 0 1] [0 0 0 1]^T)^-1 [0 0 0 1]^T

Simplifying this expression, we get the matrix representation of T as:

T = [0 0 0 0