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## Homework Statement

Let T: R4 --> R4 be defined by T(a,b,c,d) = (4a-b+c, b+3d, 3a+c+d, c-d, b+c+2d)

where a,b,c,d are in R. Compute rank(T).

## Homework Equations

## The Attempt at a Solution

Writing the transformation as a matrix,

T[a] = [4 -1 1 0][a]

**[0 1 0 3]**

[c] [3 0 1 1][c]

[d] [0 0 1 -1][d]

[0 1 1 2]

So that big matrix on top, call it A, is the standard matrix. We'll need to reduce A to echelon form in order to find the rank(T).

So this is how far I got:

A = [4 -1 1 0]

[0 1 3 0]

[3 0 2 0]

[0 0 -1 1]

[0 1 3 0]

I can't seem to reduce this any further without replacing the zero elements with nonzero elements, which is not the point. Rank(T) should be the number of nonzero rows in the matrix reduced to echelon form, right? So does that mean that rank(T) = 5? Although, I don't think A is completely reduced. Maybe my arithmetic does not work.[c] [3 0 1 1][c]

[d] [0 0 1 -1][d]

[0 1 1 2]

So that big matrix on top, call it A, is the standard matrix. We'll need to reduce A to echelon form in order to find the rank(T).

So this is how far I got:

A = [4 -1 1 0]

[0 1 3 0]

[3 0 2 0]

[0 0 -1 1]

[0 1 3 0]

I can't seem to reduce this any further without replacing the zero elements with nonzero elements, which is not the point. Rank(T) should be the number of nonzero rows in the matrix reduced to echelon form, right? So does that mean that rank(T) = 5? Although, I don't think A is completely reduced. Maybe my arithmetic does not work.