In order to successfully complete this assignment you need to participate both individually and in groups during class. If you attend class in-person then have one of the instructors check your notebook and sign you out before leaving class. If you are attending asyncronously, turn in your assignment using D2L no later than 11:59pm on the day of class. See links at the end of this document for access to the class timeline for your section.

In-Class Assignment: Fundamental Spaces

Image from: https://kevinbinz.com/2017/02/20/linear-algebra/

Agenda for today's class (80 minutes)

  1. (20 minutes) Student Check-In
  2. (20 minutes) Pre-class assignment review
  3. (20 minutes) Four Fundamental Spaces
  4. (20 minutes) Practice Example

1. Student Check-In

✅ Do This: Spend some time to brainstorm and think about ways the MTH314 instructors can help student's reduce their stress while maintaining the learning goals for the class.

2. Pre-class assignment review

3. Four Fundamental Subspaces

The four fundamental subspaces

  • Columnspace, $\mathcal{C}(A)$
  • Nullspace, $\mathcal{N}(A)$
  • Rowspaces, $R(A)$
    • All linear combinations of rows
    • All the linear combinations of the colums of $A^\top$, $\mathcal{C}(A^\top)$
  • Nullspace of $A^\top$, $\mathcal{N}(A^\top)$ (the left nullspace of $A$)

Where are these spaces for a $m\times n$ matrix $A$?

  • $\mathcal{R}(A)$ is in $R^n$
  • $\mathcal{N}(A)$ is in $R^n$
  • $\mathcal{C}(A)$ is in $R^m$
  • $\mathcal{N}(A^\top)$ is in $R^m$

Calculating basis and dimension

For $\mathcal{R}(A)$

  • If $A$ undergoes row reduction to row echelon form $B$, then $\mathcal{C}(B)\neq \mathcal{C}(A)$, but $\mathcal{R}(B) = \mathcal{R}(A)$ (or $\mathcal{C}(B^\top) = \mathcal{C}(A^\top))$
  • A basis for the rowspace of $A$ (or $B$) is the first $r$ rows of $B$
    • So we row reduce $A$ and take the pivot rows and transpose them
  • The dimension is also equal to the rank $r$

For $\mathcal{N}(A)$

  • The bases are the special solutions (one for every free variable, $n-r$)
  • The dimension is $n- r$

For $\mathcal{C}(A) = \mathcal{R}(A^\top)$

  • Apply the row reduction on the transpose $A^\top$.
  • The dimension is the rank $r$

For $\mathcal{N}(A^\top)$

  • It is also called the left nullspace, because it ends up on the left (as seen below)
  • Here we have $A^\top y = 0$
    • $y^\top(A^\top)^\top = 0^\top$
    • $y^\top A = 0^\top$
    • This is (again) the special solutions for $A^\top$ (after row reduction)
  • The dimension is $m - r$

4. Practice Example:

Consider the linear transformation defined by the following matrix $A$.

$$A = \left[ \begin{matrix} 1 & 2 & 3 & 1 \\ 1 & 1 & 2 & 1 \\ 1 & 2 & 3 & 1 \end{matrix} \right] $$ 
%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing()

✅ Question: What is the reduced row echelon form of $A$? You can use sympy.

#Put your answer to the above question here.

✅ Question: Now let's calculate the row space of $A$. Note that the row space is defined by a linear combination of the non-zero row vectors in the reduced row echelon matrix:

Put your answer to the above question here

✅ Question: What is the rank of matrix $A$? You should know the rank by inspecting the reduced row echelon form. Find a numpy or sympy function that you can use to verify your answer?

## Put code here to verify your answer.

✅ Question: Using the reduced row echelon form define the leading variables in terms of the free variables for the homogeneous equation.

Put your answer to the above question here

✅ Question: The solution to the above question defines the nullspace of $A$ (aka the Kernel). Use the sympy.nullspace function to verify your answer.

# Put your code here

✅ Question: Now let's calculate the range of $A$ (column space of $A$). Note that the range is spanned by the column vectors of $A$. Transpose $A$ and calculate the reduced row echelon form of the transposed matrix like we did above.

## Put your code here

✅ Question: The nonzero row vectors of the above solution will give a basis for the range (or $\mathcal{C}(A)$). Write the range of $A$ as a linear combination of these nonzero vectors:

Put your answer to the above question here.

✅ Question: Finally, using the reduced row echelon form for $A^\top$ define the leading variables in terms of the free variables and define the null space of $A^\top$.

Put your answer to the above question here.

Congratulations, we're done!

If you attend class in-person then have one of the instructors check your notebook and sign you out before leaving class. If you are attending remote, turn in your assignment using D2L.

Course Resources:

Written by Dirk Colbry, Michigan State University Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.