Section5.2Orthogonal complements and the matrix transpose
Activity5.2.0.1.
Let \(\vvec=\twovec{-1}2\text{.}\)
(a)
Sketch the vector \(\vvec\) and one vector that is orthogonal to it.
(b)
If a vector \(\xvec\) is orthogonal to \(\vvec\text{,}\) what do we know about the dot product \(\vvec\cdot\xvec\text{?}\)
(c)
If we write \(\xvec=\twovec xy\text{,}\) use the dot product to write an equation for the vectors orthogonal to \(\vvec\) in terms of \(x\) and \(y\text{.}\)
(d)
Use this equation to sketch the set of all vectors orthogonal to \(\vvec\text{.}\)
Activity5.2.0.2.
Section 3.5 introduced the column space \(\col(A)\) and null space \(\nul(A)\) of a matrix \(A\text{.}\) Suppose that \(A\) is a matrix and \(\xvec\) is a vector satisfying \(A\xvec=\zerovec\text{.}\)
(a)
Does \(\xvec\) belong to \(\nul(A)\) or \(\col(A)\text{?}\)
(b)
Suppose that the equation \(A\xvec=\bvec\) is consistent. Does \(\bvec\) belong to \(\nul(A)\) or \(\col(A)\text{?}\)
Definition5.2.0.1.
Given a subspace \(W\) of \(\real^m\text{,}\) the orthogonal complement of \(W\) is the set of vectors in \(\real^m\) each of which is orthogonal to every vector in \(W\text{.}\) We denote the orthogonal complement by \(W^\perp\text{.}\)
The next two activities help us find a description of the orthogonal complement \(W^\perp\text{.}\)
Activity5.2.0.3.
Suppose that \(\wvec_1=\threevec10{-2}\) and \(\wvec_2=\threevec11{-1}\) form a basis for \(W\text{,}\) a two-dimensional subspace of \(\real^3\text{.}\)
(a)
Suppose that the vector \(\xvec\) is orthogonal to \(\wvec_1\text{.}\) If we write \(\xvec=\threevec{x_1}{x_2}{x_3}\text{,}\) use the fact that \(\wvec_1\cdot\xvec =
0\) to write a linear equation for \(x_1\text{,}\)\(x_2\text{,}\) and \(x_3\text{.}\)
(b)
Suppose that \(\xvec\) is also orthogonal to \(\wvec_2\text{.}\) In the same way, write a linear equation for \(x_1\text{,}\)\(x_2\text{,}\) and \(x_3\) that arises from the fact that \(\wvec_2\cdot\xvec =
0\text{.}\)
(c)
If \(\xvec\) is orthogonal to both \(\wvec_1\) and \(\wvec_2\text{,}\) these two equations give us a linear system \(B\xvec=\zerovec\) for some matrix \(B\text{.}\) Identify the matrix \(B\) and write a parametric description of the solution space to the equation \(B\xvec = \zerovec\text{.}\)
Activity5.2.0.4.
Suppose that \(\wvec_1=\threevec10{-2}\) and \(\wvec_2=\threevec11{-1}\) form a basis for \(W\text{,}\) a two-dimensional subspace of \(\real^3\text{.}\)
(a)
Since \(\wvec_1\) and \(\wvec_2\) form a basis for the two-dimensional subspace \(W\text{,}\) any vector in \(\wvec\) in \(W\) can be written as a linear combination
If \(\xvec\) is orthogonal to both \(\wvec_1\) and \(\wvec_2\text{,}\) use the distributive property of dot products to explain why \(\xvec\) is orthogonal to \(\wvec\text{.}\)
(b)
Give a basis for the orthogonal complement \(W^\perp\) and state the dimension \(\dim W^\perp\text{.}\)
(c)
Describe \((W^\perp)^\perp\text{,}\) the orthogonal complement of \(W^\perp\text{.}\)
Definition5.2.0.2.
The transpose of the \(m\times n\) matrix \(A\) is the \(n\times m\) matrix \(A^T\) whose rows are the columns of \(A\text{.}\)
The next activity illustrates how multiplying a vector by \(A^T\) is related to computing dot products with the columns of \(A\text{.}\) You'll develop a better understanding of this relationship if you compute the dot products and matrix products in this activity without using technology.
Find the dot products \(\vvec_1\cdot\wvec\) and \(\vvec_2\cdot\wvec\text{.}\)
(b)
Now write the matrix \(A = \begin{bmatrix} \vvec_1 \amp \vvec_2 \end{bmatrix}\) and its transpose \(A^T\text{.}\) Find the product \(A^T\wvec\) and describe how this product computes both dot products \(\vvec_1\cdot\wvec\) and \(\vvec_2\cdot\wvec\text{.}\)
(c)
Suppose that \(\xvec\) is a vector that is orthogonal to both \(\vvec_1\) and \(\vvec_2\text{.}\) What does this say about the dot products \(\vvec_1\cdot\xvec\) and \(\vvec_2\cdot\xvec\text{?}\) What does this say about the product \(A^T\xvec\text{?}\)
(d)
Use the matrix \(A^T\) to give a parametric description of all the vectors \(\xvec\) that are orthogonal to \(\vvec_1\) and \(\vvec_2\text{.}\)
Activity5.2.0.7.
Remember that \(\nul(A^T)\text{,}\) the null space of \(A^T\text{,}\) is the solution set of the equation \(A^T\xvec=\zerovec\text{.}\) If \(\xvec\) is a vector in \(\nul(A^T)\text{,}\) explain why \(\xvec\) must be orthogonal to both \(\vvec_1\) and \(\vvec_2\text{.}\)
Activity5.2.0.8.
Remember that \(\col(A)\text{,}\) the column space of \(A\text{,}\) is the set of linear combinations of the columns of \(A\text{.}\) Therefore, any vector in \(\col(A)\) can be written as \(c_1\vvec_1+c_2\vvec_2\text{.}\) If \(\xvec\) is a vector in \(\nul(A^T)\text{,}\) explain why \(\xvec\) is orthogonal to every vector in \(\col(A)\text{.}\)
Activity5.2.0.9.
In Sage, the transpose of a matrix A is given by A.T. Define the matrices
Evaluate \((A+B)^T\) and \(A^T+B^T\text{.}\) What do you notice about the relationship between these two matrices?
(b)
What happens if you transpose a matrix twice; that is, what is \((A^T)^T\text{?}\)
(c)
Find \(\det(C)\) and \(\det(C^T)\text{.}\) What do you notice about the relationship between these determinants?
(d)
Find the product \(AC\) and its transpose \((AC)^T\text{.}\)
(e)
Is it possible to compute the product \(A^TC^T\text{?}\) Explain why or why not.
(f)
Find the product \(C^TA^T\) and compare it to \((AC)^T\text{.}\) What do you notice about the relationship between these two matrices?
Activity5.2.0.10.
What is the transpose of the identity matrix \(I\text{?}\)
Activity5.2.0.11.
If a square matrix \(D\) is invertible, explain why you can guarantee that \(D^T\) is invertible and why \((D^T)^{-1} = (D^{-1})^T\text{.}\)
Activity5.2.0.12.
Suppose that \(W\) is a \(5\)-dimensional subspace of \(\real^9\) and that \(A\) is a matrix whose columns form a basis for \(W\text{;}\) that is, \(\col(A) =
W\text{.}\)
(a)
What are the dimensions of \(A\text{?}\)
(b)
What is the rank of \(A\text{?}\)
(c)
What are the dimensions of \(A^T\text{?}\)
(d)
What is the rank of \(A^T\text{?}\)
(e)
What is \(\dim\nul(A^T)\text{?}\)
(f)
What is \(\dim W^\perp\text{?}\)
(g)
How are the dimensions of \(W\) and \(W^\perp\) related?
Activity5.2.0.13.
Suppose that \(W\) is a subspace of \(\real^4\) having basis
