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Section 2.2 Arithmetic Theorems
We will
learn a list of algebraic properties of vectors and matrices,
identify these by name (learned in high school)
prove a few of these properties.
Subsection 2.2.1 Algebraic Properties
In this section we want to connect properties of vectors to similar looking properties of real numbers we already know.
Checkpoint 2.2.1 .
Select a name to match each property. Be careful to distinguish between scalar and vector. For example, \(\vec{u}+\vec{v} = \vec{v}+\vec{u}\) is vector additive commutativity .
\(\displaystyle (\vec{u}+\vec{v})+\vec{w} = \vec{u}+(\vec{v}+\vec{w}). \)
\(\displaystyle \vec{u}+\vec{0} = \vec{0}+\vec{u} = \vec{u}. \)
\(\displaystyle \vec{u}+(\vec{-u}) = (\vec{-u})+\vec{u} = \vec{0}. \)
\(\displaystyle c(\vec{u}+\vec{v}) = c\vec{u}+c\vec{v}. \)
\(\displaystyle (c+d)\vec{u} = c\vec{u}+d\vec{u}. \)
\(\displaystyle c(d\vec{u}) = (cd)\vec{u}. \)
\(1\vec{u}=\vec{u}\text{.}\)
Commutative
Associative
Distributive
Inverse
Checkpoint 2.2.2 .
Select a name for each property. Be careful to distinguish between scalar and matrix.
\(\displaystyle A+B = B+A. \)
\(\displaystyle (A+B)+C = A+(B+C). \)
\(\displaystyle A+0 = 0+A = A. \)
\(\displaystyle A+(-A) = (-A)+A = 0. \)
\(\displaystyle r(A+B) = rA+rB. \)
\(\displaystyle (r+s)A = rA+sA. \)
\(\displaystyle r(sA) = (rs)A. \)
\(\displaystyle 1A = A. \)
Commutative
Associative
Distributive
Inverse
Checkpoint 2.2.3 .
Select a name for each property. Be careful to distinguish between scalar, vector, and matrix.
Commutative
Associative
Distributive
Inverse
Checkpoint 2.2.4 .
Select a name for each property. No two have the exact same description: be precise! The last item will be explained in a future lesson.
\(\displaystyle A(BC)=(AB)C. \)
\(\displaystyle A(B+C)=AB+AC. \)
\(\displaystyle (B+C)A=BA+CA. \)
\(\displaystyle r(AB)=(rA)B=A(rB). \)
\(\displaystyle I_mA=A=AI_m. \)
Commutative
Associative
Distributive
Inverse
Subsection 2.2.2 Proofs
Theorem 2.2.5 .
\(c(\vec{u}+\vec{v}) = c\vec{u}+c\vec{v}\)
Proof.
\begin{align*}
c(\vec{u}+\vec{v}) = & c([u_1, u_2, \ldots, u_n]+[v_1, v_2, \ldots, v_n]) \\
= & c([u_1+v_1, u_2+v_2, \ldots, u_n+v_n]) \\
& & \mbox{ definition of vector addition} \\
= & [c(u_1+v_1), c(u_2+v_2), \ldots, c(u_n+v_n)] \\
& & \mbox{ definition of scalar multiplication} \\
= & [cu_1+cv_1, cu_2+cv_2, \ldots, cu_n+cv_n] \\
& & \mbox{ property of real arithmetic} \\
= & [cu_1, cu_2, \ldots, cu_n]+[cv_1, cv_2, \ldots, cv_n] \\
& & \mbox{ definition of vector addition} \\
= & c[u_1, u_2, \ldots, u_n]+c[v_1, v_2, \ldots, v_n] \\
& & \mbox{ definition of scalar multiplication} \\
= & c\vec{u}+c\vec{v}.
\end{align*}
Checkpoint 2.2.6 .
Prove the following by doing the algebra steps.
\(\displaystyle \vec{u}+\vec{v} = \vec{v}+\vec{u}\)
\(\displaystyle (\vec{u}+\vec{v})+\vec{w} = \vec{u}+(\vec{v}+\vec{w}). \)
\(\displaystyle A(B+C)=AB+AC. \)
