Lab 3. Linear Spaces, Subspaces, and Linear Maps: Independent Study

This lab accompanies Chapter 3: Linear Spaces, Subspaces, Linear Transformations, Quotients, and Matrices.

The goal is to connect the abstract language of vector spaces with concrete computation:

Subspaces: zero-vector test, closure, span, and direct sums.
Linear maps: kernels, images, injectivity, surjectivity, and rank-nullity.
Quotients: collapsing a subspace to zero and counting the remaining dimensions.
Tensors: Kronecker products as a computational model of tensor products.

This is an independent-study lab. The questions include solutions so that students can check their work as they go.

Python practice notebook

You may also use the Jupyter notebook version for longer Python practice:

Interactive lab

Study guide and worked questions

Question 1. Subspace or affine set?

Determine whether each set is a subspace of the indicated vector space.

$U=\{(x,y)\in\mathbb R^2:2x-y=0\}$.
$A=\{(x,y)\in\mathbb R^2:2x-y=1\}$.
$C=\{(x,y)\in\mathbb R^2:x\ge 0, y\ge 0\}$.

Solution

$U$ is a subspace. It is the kernel of the linear functional $T(x,y)=2x-y$.
$A$ is not a subspace because $(0,0)\notin A$.
$C$ is not a subspace because it is not closed under multiplication by negative scalars. For example, $(1,1)\in C$, but $-(1,1)=(-1,-1)\notin C$.

Similar practice

Determine whether $H=\{(x,y,z)\in\mathbb R^3:x+2y-z=0\}$ is a subspace.

Answer: Yes. It is the kernel of $T(x,y,z)=x+2y-z$.

Question 2. Span and membership

Let

\[ u_1=(1,0,2),\qquad \nu_2=(0,1,-1),\]

and let $W=\operatorname{span}(\nu_1,\nu_2)$.

Describe all vectors in $W$.
Determine whether $(3,4,2)$ belongs to $W$.
Determine whether $(3,4,1)$ belongs to $W$.

Solution

A general vector in $W$ has the form

\[ s\nu_1+t\nu_2=(s,t,2s-t). \]

Thus

\[ W=\{(x,y,z)\in\mathbb R^3:z=2x-y\}. \]

For $(3,4,2)$, choose $s=3$ and $t=4$. Then $2s-t=6-4=2$, so $(3,4,2)\in W$.

For $(3,4,1)$, the first two coordinates force $s=3$ and $t=4$, but then $2s-t=2\ne 1$. Hence $(3,4,1)\notin W$.

Similar practice

For the same $W$, determine whether $(2,-1,5)$ belongs to $W$.

Answer: Yes, because $2(2)-(-1)=5$.

Question 3. Direct sums

Let

\[ U=\{(x,0,0):x\in\mathbb R\},\qquad W=\{(0,y,z):y,z\in\mathbb R\}. \]

Show that $\mathbb R^3=U\oplus W$.

Solution

Every vector $(a,b,c)\in\mathbb R^3$ can be written as

\[ (a,b,c)=(a,0,0)+(0,b,c), \]

where $(a,0,0)\in U$ and $(0,b,c)\in W$. Also,

\[ U\cap W=\{(0,0,0)\}. \]

Therefore $\mathbb R^3=U\oplus W$.

Similar practice

Let

\[ U=\{(x,y,0):x,y\in\mathbb R\},\qquad W=\{(0,y,z):y,z\in\mathbb R\}. \]

Is $\mathbb R^3=U\oplus W$?

Answer: No. Although $U+W=\mathbb R^3$, the intersection is $U\cap W=\{(0,y,0):y\in\mathbb R\}$, which is not $\{0\}$.

Question 4. Kernel and image

Let $T:\mathbb R^4\to\mathbb R^3$ be defined by $T(x)=Ax$, where

\[ A=\begin{bmatrix} 0&0&2&8\\ 1&5&2&-5\\ 2&10&6&-2 \end{bmatrix}. \]

A row reduction gives

\[ \operatorname{rref}(A)= \begin{bmatrix} 1&5&0&-13\\ 0&0&1&4\\ 0&0&0&0 \end{bmatrix}. \]

Find bases for $\operatorname{im}(T)$ and $\ker(T)$.

Solution

The pivot columns are columns $1$ and $3$ of the original matrix. Therefore

\[ \operatorname{im}(T)=\operatorname{span}\left\{ \begin{bmatrix}0\\1\\2\end{bmatrix}, \begin{bmatrix}2\\2\\6\end{bmatrix} \right\}. \]

From the RREF equations,

\[ x_1+5x_2-13x_4=0,\qquad x_3+4x_4=0. \]

Thus

\[ x_1=-5x_2+13x_4,\qquad x_3=-4x_4. \]

Let $x_2=s$ and $x_4=t$. Then

\[ x=s\begin{bmatrix}-5\\1\\0\\0\end{bmatrix} +t\begin{bmatrix}13\\0\\-4\\1\end{bmatrix}. \]

Hence

\[ \ker(T)=\operatorname{span}\left\{ \begin{bmatrix}-5\\1\\0\\0\end{bmatrix}, \begin{bmatrix}13\\0\\-4\\1\end{bmatrix} \right\}. \]

Rank-nullity also checks the answer:

\[ \dim\operatorname{im}(T)+\dim\ker(T)=2+2=4. \]

Similar practice

Let

\[ B=\begin{bmatrix} 1&0&0&2\\ 0&1&0&-1\\ 0&0&1&3 \end{bmatrix}. \]

Find $\ker(B)$.

Answer: The equations are $x_1+2x_4=0$, $x_2-x_4=0$, $x_3+3x_4=0$. Let $x_4=t$. Then

\[ \ker(B)=\operatorname{span}\left\{\begin{bmatrix}-2\\1\\-3\\1\end{bmatrix}\right\}. \]

Question 5. Quotient dimension

Let $N$ be a 2-dimensional subspace of $\mathbb R^4$. What is $\dim(\mathbb R^4/N)$?

Solution

By the quotient dimension formula,

\[ \dim(\mathbb R^4/N)=\dim(\mathbb R^4)-\dim(N)=4-2=2. \]

Similar practice

Let $N=\operatorname{span}\{(1,0,0),(0,1,0)\}\subseteq\mathbb R^3$. What is $\dim(\mathbb R^3/N)$?

Answer: $3-2=1$.

Question 6. Kronecker product

Let

\[ A=\begin{bmatrix}1&2\\3&4\end{bmatrix},\qquad B=\begin{bmatrix}0&5\\6&7\end{bmatrix}. \]

Compute $A\otimes B$.

Solution

By definition,

\[ A\otimes B= \begin{bmatrix} 1B&2B\\ 3B&4B \end{bmatrix}. \]

Therefore

\[ A\otimes B= \begin{bmatrix} 0&5&0&10\\ 6&7&12&14\\ 0&15&0&20\\ 18&21&24&28 \end{bmatrix}. \]

Question 7. AI companion critique

Ask an AI tool:

Explain the difference between a vector space, a subspace, and a quotient space using examples from $\mathbb R^3$.

Then critique the response.

Your critique should answer:

Does it correctly state that a subspace must contain the zero vector?
Does it distinguish a plane through the origin from an affine plane?
Does it explain that a quotient space identifies vectors differing by a subspace?
Does it include a dimension example such as $\dim(\mathbb R^3/N)=3-\dim(N)$?

# Lab 3. Linear Spaces, Subspaces, and Linear Maps: Independent Study {.unnumbered} This lab accompanies **Chapter 3: Linear Spaces, Subspaces, Linear Transformations, Quotients, and Matrices**. The goal is to connect the abstract language of vector spaces with concrete computation: 1. **Subspaces:** zero-vector test, closure, span, and direct sums. 2. **Linear maps:** kernels, images, injectivity, surjectivity, and rank-nullity. 3. **Quotients:** collapsing a subspace to zero and counting the remaining dimensions. 4. **Tensors:** Kronecker products as a computational model of tensor products. This is an **independent-study lab**. The questions include solutions so that students can check their work as they go. ## Python practice notebook You may also use the Jupyter notebook version for longer Python practice: - [Download Lab 3 independent-study notebook](lab-03-linear-spaces-subspaces-maps-independent-study.ipynb) - [Open Lab 3 in Google Colab](https://colab.research.google.com/github/wanghemath/MATH5110/blob/main/Labs/lab-03-linear-spaces-subspaces-maps-independent-study.ipynb) ## Interactive lab ```{=html} <iframe src="lab-03-interactive.html" width="100%" height="2600" style="border:1px solid #ddd; border-radius:12px;"> </iframe> ``` ## Study guide and worked questions ### Question 1. Subspace or affine set? Determine whether each set is a subspace of the indicated vector space. 1. $U=\{(x,y)\in\mathbb R^2:2x-y=0\}$. 2. $A=\{(x,y)\in\mathbb R^2:2x-y=1\}$. 3. $C=\{(x,y)\in\mathbb R^2:x\ge 0, y\ge 0\}$. #### Solution 1. $U$ is a subspace. It is the kernel of the linear functional $T(x,y)=2x-y$. 2. $A$ is not a subspace because $(0,0)\notin A$. 3. $C$ is not a subspace because it is not closed under multiplication by negative scalars. For example, $(1,1)\in C$, but $-(1,1)=(-1,-1)\notin C$. #### Similar practice Determine whether $H=\{(x,y,z)\in\mathbb R^3:x+2y-z=0\}$ is a subspace. **Answer:** Yes. It is the kernel of $T(x,y,z)=x+2y-z$. ### Question 2. Span and membership Let $$ u_1=(1,0,2),\qquad \nu_2=(0,1,-1),$$ and let $W=\operatorname{span}(\nu_1,\nu_2)$. 1. Describe all vectors in $W$. 2. Determine whether $(3,4,2)$ belongs to $W$. 3. Determine whether $(3,4,1)$ belongs to $W$. #### Solution A general vector in $W$ has the form $$ s\nu_1+t\nu_2=(s,t,2s-t). $$ Thus $$ W=\{(x,y,z)\in\mathbb R^3:z=2x-y\}. $$ For $(3,4,2)$, choose $s=3$ and $t=4$. Then $2s-t=6-4=2$, so $(3,4,2)\in W$. For $(3,4,1)$, the first two coordinates force $s=3$ and $t=4$, but then $2s-t=2\ne 1$. Hence $(3,4,1)\notin W$. #### Similar practice For the same $W$, determine whether $(2,-1,5)$ belongs to $W$. **Answer:** Yes, because $2(2)-(-1)=5$. ### Question 3. Direct sums Let $$ U=\{(x,0,0):x\in\mathbb R\},\qquad W=\{(0,y,z):y,z\in\mathbb R\}. $$ Show that $\mathbb R^3=U\oplus W$. #### Solution Every vector $(a,b,c)\in\mathbb R^3$ can be written as $$ (a,b,c)=(a,0,0)+(0,b,c), $$ where $(a,0,0)\in U$ and $(0,b,c)\in W$. Also, $$ U\cap W=\{(0,0,0)\}. $$ Therefore $\mathbb R^3=U\oplus W$. #### Similar practice Let $$ U=\{(x,y,0):x,y\in\mathbb R\},\qquad W=\{(0,y,z):y,z\in\mathbb R\}. $$ Is $\mathbb R^3=U\oplus W$? **Answer:** No. Although $U+W=\mathbb R^3$, the intersection is $U\cap W=\{(0,y,0):y\in\mathbb R\}$, which is not $\{0\}$. ### Question 4. Kernel and image Let $T:\mathbb R^4\to\mathbb R^3$ be defined by $T(x)=Ax$, where $$ A=\begin{bmatrix} 0&0&2&8\\ 1&5&2&-5\\ 2&10&6&-2 \end{bmatrix}. $$ A row reduction gives $$ \operatorname{rref}(A)= \begin{bmatrix} 1&5&0&-13\\ 0&0&1&4\\ 0&0&0&0 \end{bmatrix}. $$ Find bases for $\operatorname{im}(T)$ and $\ker(T)$. #### Solution The pivot columns are columns $1$ and $3$ of the original matrix. Therefore $$ \operatorname{im}(T)=\operatorname{span}\left\{ \begin{bmatrix}0\\1\\2\end{bmatrix}, \begin{bmatrix}2\\2\\6\end{bmatrix} \right\}. $$ From the RREF equations, $$ x_1+5x_2-13x_4=0,\qquad x_3+4x_4=0. $$ Thus $$ x_1=-5x_2+13x_4,\qquad x_3=-4x_4. $$ Let $x_2=s$ and $x_4=t$. Then $$ x=s\begin{bmatrix}-5\\1\\0\\0\end{bmatrix} +t\begin{bmatrix}13\\0\\-4\\1\end{bmatrix}. $$ Hence $$ \ker(T)=\operatorname{span}\left\{ \begin{bmatrix}-5\\1\\0\\0\end{bmatrix}, \begin{bmatrix}13\\0\\-4\\1\end{bmatrix} \right\}. $$ Rank-nullity also checks the answer: $$ \dim\operatorname{im}(T)+\dim\ker(T)=2+2=4. $$ #### Similar practice Let $$ B=\begin{bmatrix} 1&0&0&2\\ 0&1&0&-1\\ 0&0&1&3 \end{bmatrix}. $$ Find $\ker(B)$. **Answer:** The equations are $x_1+2x_4=0$, $x_2-x_4=0$, $x_3+3x_4=0$. Let $x_4=t$. Then $$ \ker(B)=\operatorname{span}\left\{\begin{bmatrix}-2\\1\\-3\\1\end{bmatrix}\right\}. $$ ### Question 5. Quotient dimension Let $N$ be a 2-dimensional subspace of $\mathbb R^4$. What is $\dim(\mathbb R^4/N)$? #### Solution By the quotient dimension formula, $$ \dim(\mathbb R^4/N)=\dim(\mathbb R^4)-\dim(N)=4-2=2. $$ #### Similar practice Let $N=\operatorname{span}\{(1,0,0),(0,1,0)\}\subseteq\mathbb R^3$. What is $\dim(\mathbb R^3/N)$? **Answer:** $3-2=1$. ### Question 6. Kronecker product Let $$ A=\begin{bmatrix}1&2\\3&4\end{bmatrix},\qquad B=\begin{bmatrix}0&5\\6&7\end{bmatrix}. $$ Compute $A\otimes B$. #### Solution By definition, $$ A\otimes B= \begin{bmatrix} 1B&2B\\ 3B&4B \end{bmatrix}. $$ Therefore $$ A\otimes B= \begin{bmatrix} 0&5&0&10\\ 6&7&12&14\\ 0&15&0&20\\ 18&21&24&28 \end{bmatrix}. $$ ### Question 7. AI companion critique Ask an AI tool: > Explain the difference between a vector space, a subspace, and a quotient space using examples from $\mathbb R^3$. Then critique the response. Your critique should answer: 1. Does it correctly state that a subspace must contain the zero vector? 2. Does it distinguish a plane through the origin from an affine plane? 3. Does it explain that a quotient space identifies vectors differing by a subspace? 4. Does it include a dimension example such as $\dim(\mathbb R^3/N)=3-\dim(N)$?