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Agenda for today's class (80 minutes)¶
Consider the following example. We claim that the following set of vectors form a baiss for $R^3$:
$$B = \{(2,1, 3), (-1,6, 0), (3, 4, -10) \}$$If these vectors form a basis they must be linearly independent and Span the entire space of $R^3$
%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True)
✅ **DO THIS:** Create a $3 \times 3$ numpy matrix $A$ where the columns of $A$ form are the basis vectors.
#Put your answer to the above question here
from answercheck import checkanswer
checkanswer.matrix(A,'68b81f1c1041158b519936cb1a2e4d6b');
✅ **DO THIS:** Using python, Calculate the determinant of matrix $A$?
# Put your answer to the above question here.
✅ **DO THIS:** Using python, calculate the inverse of $A$.
# Put your answer to the above question here.
✅ **DO THIS:** Using python, calculate the rank of $A$.
# Put your answer to the above question here.
✅ **DO THIS:** Using python, calculate the reduced row echelon form of $A$.
# Put your answer to the above question here.
✅ **DO THIS:** Using the above $A$ and the vector $b=(1,3,2)$. What is the solution to $Ax=b$?
#Put your answer to the above question here.
from answercheck import checkanswer
checkanswer.matrix(x,'8b0938260dfeaafc9f8e9fec0bc72f17');
Turns out a matrix where column vectors are formed from basis vectors a lot of interesting properties and the following statements are equivalent.
- The column vectors of $A$ form a basis for $R^n$
- $|A| \ne 0$
- $A$ is invertible.
- $A$ is row equivalent to $I_n$ (i.e. it's reduced row echelon form is $I_n$)
- The system of equations $Ax = b$ has a unique solution.
- $rank(A) = n$
Not all matrices follow the above statements but the ones that do are used throughout linear algebra so it is important that we know these properties.
3. Vector Spaces¶
A Vector Space is a set $V$ of elements called vectors, having operations of addition and scalar multiplication defined on it that satisfy the following conditions ($u$, $v$, and $w$ are arbitrary elements of $V$, and c and d are scalars.)
Closure Axioms¶
- The sum $u + v$ exists and is an element of $V$. ($V$ is closed under addition.)
- $cu$ is an element of $V$. ($V$ is closed under scalar multiplication.)
Addition Axioms¶
- $u + v = v + u$ (commutative property)
- $u + (v + w) = (u + v) + w$ (associative property)
- There exists an element of $V$, called a zero vector, denoted $0$, such that $u+0 = u$
- For every element $u$ of $V$, there exists an element called a negative of $u$, denoted $-u$, such that $u + (-u) = 0$.
Scalar Multiplication Axioms¶
- $c(u+v) = cu + cv$
- $(c + d)u = cu + du$
- $c(du) = (cd)u$
- $1u = u$
Definition of a basis of a vector space¶
A finite set of vectors ${v_1,\dots, v_n}$ is called a basis of a vector space $V$ if the set spans $V$ and is linearly independent. i.e. each vector in $V$ can be expressed uniquely as a linear combination of the vectors in a basis.
Vector spaces¶
✅ **DO THIS:** Let $U$ be the set of all circles in $R^2$ having center at the origin. Interpret the origin as being in this set, i.e., it is a circle center at the origin with radius zero. Assume $C_1$ and $C_2$ are elements of $U$. Let $C_1 + C_2$ be the circle centered at the origin, whose radius is the sum of the radii of $C_1$ and $C_2$. Let $kC_1$ be the circle center at the origin, whose radius is $|k|$ times that of $C_1$. Determine which vector space axioms hold and which do not. (See page 216 of the textbook).
Put your answer here
Spans:¶
✅ **DO THIS:** Let $v$, $v_1$, and $v_2$ be vectors in a vector space $V$. Let $v$ be a linear combination of $v_1$ and $v_2$. If $c_1$ and $c_2$ are nonzero real numbers, show that $v$ is also a linear combination of $c_1v_1$ and $c_2v_2$.
Put your answer here
✅ **DO THIS:** Let $v_1$ and $v_2$ span a vector space $V$. Let $v_3$ be any other vector in $V$. Show that $v_1$, $v_2$, and $v_3$ also span $V$.
Put your answer here
Linear Independent:¶
Read the definition for Linear Dependent and Linearly independent in the textbook and consider the following matrix, which is in the reduced row echelon form.
$$ \left[ \begin{matrix} 1 & 0 & 0 & 7 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 3 \end{matrix} \right] $$✅ **DO THIS:** Show that the row vectors form a linearly independent set:
Put your answer here
✅ **DO THIS:** Is the set of nonzero row vectors of any matrix in reduced row echelon form linearly independent? Discuss in your groups and include your thoughts below.
Put your answer here
✅ **DO THIS:** A computer program accepts a number of vectors in $R^3$ as input and checks to see if the vectors are linearly independent and outputs a True/False statment. Discuss in your groups, which is more likely to happen due to round-off error--that the computer states that a given set of linearly independent vectors is linearly dependent, or vice versa? Put your groups thoughts below.
Put your answer here
Congratulations, we're done!¶
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Course Resources:¶
Written by Dirk Colbry, Michigan State University
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.