Linear Algebra for Team-Based Inquiry Learning
2022 Edition
Steven Clontz | Drew Lewis |
---|---|
University of South Alabama | University of South Alabama |
August 2, 2022
Section 4.1: Matrices and Multiplication (MX1)
Observation 4.1.1
If T: \IR^n \rightarrow \IR^m and S: \IR^m \rightarrow \IR^k are linear maps, then the composition map S\circ T is a linear map from \IR^n \rightarrow \IR^k\text{.}
Recall that for a vector, \vec{v} \in \IR^n\text{,} the composition is computed as (S \circ T)(\vec{v})=S(T(\vec{v}))\text{.}
Activity 4.1.2 (~5 min)
Let T: \IR^3 \rightarrow \IR^2 be given by the 2\times 3 standard matrix B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right] and S: \IR^2 \rightarrow \IR^4 be given by the 4\times 2 standard matrix A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}
What are the domain and codomain of the composition map S \circ T\text{?}
The domain is \IR ^3 and the codomain is \IR^2
The domain is \IR ^2 and the codomain is \IR^4
The domain is \IR ^3 and the codomain is \IR^4
The domain is \IR ^4 and the codomain is \IR^3
Activity 4.1.3 (~2 min)
Let T: \IR^3 \rightarrow \IR^2 be given by the 2\times 3 standard matrix B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right] and S: \IR^2 \rightarrow \IR^4 be given by the 4\times 2 standard matrix A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}
What size will the standard matrix of S \circ T:\IR^3\to\IR^4 be? (Rows \times Columns)
- \displaystyle 4 \times 3
- \displaystyle 3 \times 4
- \displaystyle 3 \times 2
- \displaystyle 2 \times 4
Activity 4.1.4 (~15 min)
Let T: \IR^3 \rightarrow \IR^2 be given by the 2\times 3 standard matrix B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right] and S: \IR^2 \rightarrow \IR^4 be given by the 4\times 2 standard matrix A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}
Part 1.
Compute
Activity 4.1.4 (~15 min)
Let T: \IR^3 \rightarrow \IR^2 be given by the 2\times 3 standard matrix B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right] and S: \IR^2 \rightarrow \IR^4 be given by the 4\times 2 standard matrix A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}
Part 2.
Compute (S \circ T)(\vec{e}_2) \text{.}
Activity 4.1.4 (~15 min)
Let T: \IR^3 \rightarrow \IR^2 be given by the 2\times 3 standard matrix B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right] and S: \IR^2 \rightarrow \IR^4 be given by the 4\times 2 standard matrix A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}
Part 3.
Compute (S \circ T)(\vec{e}_3) \text{.}
Activity 4.1.4 (~15 min)
Let T: \IR^3 \rightarrow \IR^2 be given by the 2\times 3 standard matrix B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right] and S: \IR^2 \rightarrow \IR^4 be given by the 4\times 2 standard matrix A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}
Part 4.
Write the 4\times 3 standard matrix of S \circ T:\IR^3\to\IR^4\text{.}
Definition 4.1.5
We define the product AB of a m \times n matrix A and a n \times k matrix B to be the m \times k standard matrix of the composition map of the two corresponding linear functions.
For the previous activity, T was a map \IR^3 \rightarrow \IR^2\text{,} and S was a map \IR^2 \rightarrow \IR^4\text{,} so S \circ T gave a map \IR^3 \rightarrow \IR^4 with a 4\times 3 standard matrix:
Activity 4.1.6 (~15 min)
Let S: \IR^3 \rightarrow \IR^2 be given by the matrix A=\left[\begin{array}{ccc} -4 & -2 & 3 \\ 0 & 1 & 1 \end{array}\right] and T: \IR^2 \rightarrow \IR^3 be given by the matrix B=\left[\begin{array}{cc} 2 & 3 \\ 1 & -1 \\ 0 & -1 \end{array}\right]\text{.}
Part 1.
Write the dimensions (rows \times columns) for A\text{,} B\text{,} AB\text{,} and BA\text{.}
Activity 4.1.6 (~15 min)
Let S: \IR^3 \rightarrow \IR^2 be given by the matrix A=\left[\begin{array}{ccc} -4 & -2 & 3 \\ 0 & 1 & 1 \end{array}\right] and T: \IR^2 \rightarrow \IR^3 be given by the matrix B=\left[\begin{array}{cc} 2 & 3 \\ 1 & -1 \\ 0 & -1 \end{array}\right]\text{.}
Part 2.
Find the standard matrix AB of S \circ T\text{.}
Activity 4.1.6 (~15 min)
Let S: \IR^3 \rightarrow \IR^2 be given by the matrix A=\left[\begin{array}{ccc} -4 & -2 & 3 \\ 0 & 1 & 1 \end{array}\right] and T: \IR^2 \rightarrow \IR^3 be given by the matrix B=\left[\begin{array}{cc} 2 & 3 \\ 1 & -1 \\ 0 & -1 \end{array}\right]\text{.}
Part 3.
Find the standard matrix BA of T \circ S\text{.}
Activity 4.1.7 (~10 min)
Consider the following three matrices.
Part 1.
Find the domain and codomain of each of the three linear maps corresponding to A\text{,} B\text{,} and C\text{.}
Activity 4.1.7 (~10 min)
Consider the following three matrices.
Part 2.
Only one of the matrix products AB,AC,BA,BC,CA,CB can actually be computed. Compute it.
Activity 4.1.8 (~15 min)
Let B=\left[\begin{array}{ccc} 3 & -4 & 0 \\ 2 & 0 & -1 \\ 0 & -3 & 3 \end{array}\right]\text{,} and let A=\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]\text{.}
Part 1.
Compute the product BA by hand.
Activity 4.1.8 (~15 min)
Let B=\left[\begin{array}{ccc} 3 & -4 & 0 \\ 2 & 0 & -1 \\ 0 & -3 & 3 \end{array}\right]\text{,} and let A=\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]\text{.}
Part 2.
Check your work using technology. Using Octave:
B = [3 -4 0 ; 2 0 -1 ; 0 -3 3] A = [2 7 -1 ; 0 3 2 ; 1 1 -1] B*A
Activity 4.1.9
Of the following three matrices, only two may be multiplied.
Activity 4.1.9
Answer.