Orthogonal Complements of Linear Transformations with Matrix A

[LosTacos]

Homework Statement

Let L: ℝn→ℝm be a linear transformation with matrix A ( with respect to the standard basis). Show that ker(L) is the orthogonal complement of the row space of A.

Homework Equations

The Attempt at a Solution

The ker(L) is the subset of all vectors of V that map to 0. The orthogonal complement of W is the set of all vectors x with property that xw= 0. Would we use the dimension theorem:
dim(ker(L)) = n - dim(range(L)) = n - dim(W) =dim(Wτ). Since Wτ is contained in ker(L).

*Wτ is orthogonal complement.

 

[LosTacos]

This seems way to obvious. The row space is the subset of all vectors that are linear combinations of rows of A. So the kernel of L is the subset of all vectors in V that map to 0. Then, the orthogonal complement is the set of vectors with the property that x ⋅ w = 0. So, I need to prove that the vector x in the orthogonal complement is the same vector in the row space?

 

[Mark44]

Homework Statement

Let L: ℝn→ℝm be a linear transformation with matrix A ( with respect to the standard basis). Show that ker(L) is the orthogonal complement of the row space of A.

Homework Equations

The Attempt at a Solution

The ker(L) is the subset of all vectors of V that map to 0. The orthogonal complement of W is the set of all vectors x with property that xw= 0. Would we use the dimension theorem:
dim(ker(L)) = n - dim(range(L)) = n - dim(W) =dim(Wτ). Since Wτ is contained in ker(L).

*Wτ is orthogonal complement.

Let's use W in a meaningful way since you haven't defined what W represents. Let W = ker(L).

The orthogonal complement of W is the set of all vectors x with property that x $\cdot$ w= 0, where w $\in$ W.

Now, assume that x is any vector in the row space of L. How can you write x, knowing what you know about L and its matrix representation?

 

[LosTacos]

If a is scalars, then ax₁ + ax₂ + ... + ax_k = 0

 

[Mark44]

If a is scalars, then ax₁ + ax₂ + ... + ax_k = 0

What does this equation mean? I have no idea what you're doing.

 

[LosTacos]

That was the linear combination of the row space. That was how I thought to write x in terms of L. And it's equal to 0 because it is the kernel.

 

[Mark44]

That was the linear combination of the row space. That was how I thought to write x in terms of L. And it's equal to 0 because it is the kernel.

There are several things wrong with this.

1. There are not k rows in the matrix, so it makes no sense to list them as x₁, x₂, ... , x_k.
2. The row space is not in the kernel. Your goal in this problem is to show that the row space is the orthogonal complement of the kernel.
3. A vector in the kernel doesn't have to be 0.

 

[LosTacos]

Okay, given matrix A and vector x:

A⋅x = 0 means that w_k⋅x = 0 for ever row vector w_k in R. Therefore, the orthogonal complement of row space is kernel.

 

[Mark44]

Okay, given matrix A and vector x:

Any old vector x? Is x in the row space of A or is it in the kernel of A?

A⋅x = 0 means that w_k⋅x = 0 for ever row vector w_k in R.

I don't follow this. Ax = 0 means only that x is in the kernel of A. Why does it follow that w_k⋅x = 0? Neither w_k or x has to be the zero vector.

Therefore, the orthogonal complement of row space is kernel.