TBIL Activities for Understanding Linear Algebra

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
\begin{equation*} \wvec = c_1\wvec_1 + c_2\wvec_2\text{.} \end{equation*}
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 $$A\text{.}$$

Activity5.2.0.5.

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

Activity5.2.0.6.

Suppose that
\begin{equation*} \vvec_1=\threevec20{-2},\hspace{24pt} \vvec_2=\threevec112,\hspace{24pt} \wvec=\threevec{-2}23\text{.} \end{equation*}

(a)

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
\begin{equation*} A = \begin{bmatrix} 1 \amp 0 \amp -3 \\ 2 \amp -2 \amp 1 \\ \end{bmatrix}, \hspace{6pt} B = \begin{bmatrix} 3 \amp -4 \amp 1 \\ 0 \amp 1 \amp 2 \\ \end{bmatrix}, \hspace{6pt} C= \begin{bmatrix} 1 \amp 0 \amp -3 \\ 2 \amp -2 \amp 1 \\ 3 \amp 2 \amp 0 \\ \end{bmatrix}\text{.} \end{equation*}

(a)

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
\begin{equation*} \wvec_1 = \fourvec102{-1},\hspace{24pt} \wvec_2 = \fourvec{-1}2{-6}3. \end{equation*}

(a)

Find the dimensions $$\dim W$$ and $$\dim W^\perp\text{.}$$

(b)

Find a basis for $$W^\perp\text{.}$$ It may be helpful to know that the Sage command A.right_kernel() produces a basis for $$\nul(A)\text{.}$$



(c)

Verify that each of the basis vectors you found for $$W^\perp$$ are orthogonal to the basis vectors for $$W\text{.}$$