Orthogonal complements and the matrix transpose

Suppose that $\wvec_1=\threevec10{-2}$ and $\wvec_2=\threevec11{-1}$ form a basis for $W$, a two-dimensional subspace of $\real^3$. We will find a description of the orthogonal complement $W^\perp$.

1. Suppose that the vector $\xvec$ is orthogonal to $\wvec_1$. If we write $\xvec=\threevec{x_1}{x_2}{x_3}$, use the fact that $\wvec_1\cdot\xvec = 0$ to write a linear equation for $x_1$, $x_2$, and $x_3$.

2. Suppose that $\xvec$ is also orthogonal to $\wvec_2$. In the same way, write a linear equation for $x_1$, $x_2$, and $x_3$ that arises from the fact that $\wvec_2\cdot\xvec = 0$.

3. If $\xvec$ is orthog…

This activity illustrates how multiplying a vector by $A^T$ is related to computing dot products with the columns of $A$. You'll develop a better understanding of this relationship if you compute the dot products and matrix products in this activity without using technology.

1. If $B = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & -2 \\ \end{bmatrix}$, write the matrix $B^T$.

2. Suppose that Find the dot products $\vvec_1\cdot\wvec$ and $\vvec_2\cdot\wvec$.

3. Now write the matrix $A = \begin{bmatrix} \vvec_1 & \vvec_2 \end{bmatrix}$ and its transpose $A^T$. Find the product $A^T\wvec$…

In Sage, the transpose of a matrix A is given by A.T. Define the matrices

1. Evaluate $(A+B)^T$ and $A^T+B^T$. What do you notice about the relationship between these two matrices?

2. What happens if you transpose a matrix twice; that is, what is $(A^T)^T$?

3. Find $\det(C)$ and $\det(C^T)$. What do you notice about the relationship between these determinants?

4. 1. Find the product $AC$ and its transpose $(AC)^T$.

   2. Is it possible to compute the product $A^TC^T$? Explain why…

1. Suppose that $W$ is a $5$-dimensional subspace of $\real^9$ and that $A$ is a matrix whose columns form a basis for $W$; that is, $\col(A) = W$.

   1. What is the shape of $A$?

   2. What is the rank of $A$?

   3. What is the shape of $A^T$?

   4. What is the rank of $A^T$?

   5. What is $\dim\nul(A^T)$?

   6. What is $\dim W^\perp$?

   7. How are the dimensions of $W$ and $W^\perp$ related?

2. Suppose that $W$ is a subspace of $\real^4$ having basis

   1. Find the dimensions $\dim W$ and $\dim W^\perp$.

   2. Find a basis for $W^\perp$. It may be helpful to know that the Sa…

Suppose that $W$ is a subspace of $\real^4$ with basis

1. What are the dimensions $\dim W$ and $\dim W^\perp$?

2. Find a basis for $W^\perp$.

3. Verify that each of the basis vectors for $W^\perp$ are orthogonal to $\wvec_1$ and $\wvec_2$.

Consider the matrix $A = \begin{bmatrix} -1 & -2 & -2 \\ 1 & 3 & 4 \\ 2 & 1 & -2 \end{bmatrix}$.

1. Find $\rank(A)$ and a basis for $\col(A)$.

2. Determine the dimension of $\col(A)^\perp$ and find a basis for it.

Suppose that $W$ is the subspace of $\real^4$ defined as the solution set of the equation

1. What are the dimensions $\dim W$ and $\dim W^\perp$?

2. Find a basis for $W$.

3. Find a basis for $W^\perp$.

4. In general, how can you easily find a basis for $W^\perp$ when $W$ is defined by

Determine whether the following statements are true or false and explain your reasoning.

1. If $A=\begin{bmatrix} 2 & 1 \\ 1 & 1 \\ -3 & 1 \end{bmatrix}$, then $\xvec=\threevec{4}{-5}1$ is in $\col(A)^\perp$.

2. If $A$ is a $2\times3$ matrix and $B$ is a $3\times4$ matrix, then $(AB)^T = A^TB^T$ is a $4\times2$ matrix.

3. If the columns of $A$ are $\vvec_1$, $\vvec_2$, and $\vvec_3$ and $A^T\xvec = \threevec{2}01$, then $\xvec$ is orthogonal to $\vvec_2$.

4. If $A$ is a $4\times 4$ matrix with $\rank(A) = 3$, then $\col(A)^\perp$ is a line in $\real^4$.

5. If $A$ is a $5\times 7$ matrix with $\rank(A) = 5$, then $\rank(A^T) = 7$.

Apply properties of matrix operations to simplify the following expressions.

1. $A^T(BA^T)^{-1}$

2. $(A+B)^T(A+B)$

3. $[A(A+B)^T]^T$

4. $(A + 2I)^T$

A symmetric matrix $A$ is one for which $A=A^T$.

1. Explain why a symmetric matrix must be square.

2. If $A$ and $B$ are general matrices and $D$ is a square diagonal matrix, which of the following matrices can you guarantee are symmetric?

   1. $D$

   2. $BAB^{-1}$

   3. $AA^T$.

   4. $BDB^T$

If $A$ is a square matrix, remember that the characteristic polynomial of $A$ is $\det(A-\lambda I)$ and that the roots of the characteristic polynomial are the eigenvalues of $A$.

1. Explain why $A$ and $A^T$ have the same characteristic polynomial.

2. Explain why $A$ and $A^T$ have the same set of eigenvalues.

3. Suppose that $A$ is diagonalizable with diagonalization $A=PDP^{-1}$. Explain why $A^T$ is diagonalizable and find a diagonalization.

This exercise introduces a version of the Pythagorean theorem that we'll use later.

1. Suppose that $\vvec$ and $\wvec$ are orthogonal to one another. Use the dot product to explain why

2. Suppose that $W$ is a subspace of $\real^m$ and that $\zvec$ is a vector in $\real^m$ for which where $\xvec$ is in $W$ and $\yvec$ is in $W^\perp$. Explain why which is an expression of the Pythagorean theorem.

In the next chapter, symmetric matrices---that is, matrices for which $A=A^T$---play an important role. It turns out that eigenvectors of a symmetric matrix that are associated to different eigenvalues are orthogonal. We will explain this fact in this exercise.

1. Viewing a vector as a matrix having one column, we may write $\xvec\cdot\yvec = \xvec^T\yvec$. If $A$ is a matrix, explain why $\xvec\cdot (A\yvec) = (A^T\xvec) \cdot \yvec$.

2. We have seen that the matrix $A=\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ has eigenvectors $\vvec_1=\twovec11$…