AE 15: Linear algebra
Suggested answers
In this exercise, we will:
- Transpose a Vector: Learn how to transpose a given vector.
- Transpose a Matrix: Understand the concept of matrix transposition by working with a given matrix.
- Determine Matrix Dimensions: Identify and write down the dimensions of given matrices.
- Matrix Multiplication Validity: Check the validity of matrix multiplication for given matrices and determine the dimensions of the resulting matrix if valid.
- Compute Matrix Product: Perform the actual computation of a matrix product for given matrices when the multiplication is valid.
Linear algebra
For each exercise, show all steps clearly.
Transposition
Exercise 1
Given a vector\(\mathbf{y}\):
\[ \mathbf{y} = \begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix} \]
Write down its transpose \(\mathbf{y}^\top\)
Solution:
- The transpose of the vector \(\mathbf{y}\) is:
\[ \mathbf{y}^\top = \begin{bmatrix} y_1 & y_2 & y_3 \end{bmatrix} \]
Exercise 2
Given the following matrix \(\mathbf{N}\):
\[ \mathbf{N} = \begin{bmatrix}n_{11} & n_{12} \\n_{21} & n_{22} \\n_{31} & n_{32}\end{bmatrix} \]
Write down its transpose, \(\mathbf{N}^{\top}\)
Solution:
- The transpose of the matrix \(\mathbf{N}\) is:
\[ \mathbf{N}^\top = \begin{bmatrix}n_{11} & n_{21} & n_{31} \\n_{12} & n_{22} & n_{32}\end{bmatrix} \]
Matrix operations
Exercise 3
Consider the following matrices \(\mathbf{C}\) and \(\mathbf{D}\):
\[ \mathbf{C} = \begin{bmatrix}c_{11} & c_{12} & c_{13} \\c_{21} & c_{22} & c_{23}\end{bmatrix}, \quad\mathbf{D} = \begin{bmatrix}d_{11} & d_{12} \\d_{21} & d_{22} \\d_{31} & d_{32}\end{bmatrix} \]
What are the dimensions of \(\mathbf{C}\)?
What are the dimensions of \(\mathbf{D}\)?
For the matrix product \(\mathbf{C} \mathbf{D}\):
Determine if the product is valid, and explain why.
If the product is valid, write down the dimensions of the resulting matrix without computing the product.
Solution:
The dimensions of \(\mathbf{C}\) are \(2 \times 3\)
The dimensions of \(\mathbf{D}\) are \(3 \times 2\).
For the matrix product \(\mathbf{C} \mathbf{D}\):
The product is valid because the number of columns in \(\mathbf{C}\) (which is 3) matches the number of rows in \(\mathbf{D}\) (which is 3).
The dimensions of the resulting matrix will be \(2 \times 2\).
Exercise 4
Consider the following matrices \(\mathbf{E}\) and \(\mathbf{F}\):
\[ \mathbf{E} = \begin{bmatrix}e_{11} & e_{12} & e_{13} & e_{14} \\e_{21} & e_{22} & e_{23} & e_{24}\end{bmatrix}, \quad\mathbf{F} = \begin{bmatrix}f_{11} & f_{12} \\f_{21} & f_{22} \\f_{31} & f_{32} \\f_{41} & f_{42}\end{bmatrix} \]
What are the dimensions of \(\mathbf{E}\)?
What are the dimensions of \(\mathbf{F}\)?
For the matrix product \(\mathbf{E} \mathbf{F}\):
Determine if the product is valid, and explain why.
If the product is valid, write down the dimensions of the resulting matrix without computing the product.
Solution:
The dimensions of \(\mathbf{F}\) are \(2 \times 4\)
The dimensions of \(\mathbf{F}\) are \(4 \times 2\).
For the matrix product \(\mathbf{E} \mathbf{F}\):
The product is valid because the number of columns in \(\mathbf{E}\) (which is 4) matches the number of rows in \(\mathbf{F}\) (which is 4).
The resulting matrix is:
\[ \mathbf{E} \mathbf{F} = \begin{bmatrix}e_{11} & e_{12} & e_{13} & e_{14} \\e_{21} & e_{22} & e_{23} & e_{24}\end{bmatrix}\begin{bmatrix}f_{11} & f_{12} \\f_{21} & f_{22} \\f_{31} & f_{32} \\f_{41} & f_{42}\end{bmatrix}= \\\begin{bmatrix}e_{11}f_{11} + e_{12}f_{21} + e_{13}f_{31} + e_{14}f_{41} & e_{11}f_{12} + e_{12}f_{22} + e_{13}f_{32} + e_{14}f_{42} \\e_{21}f_{11} + e_{22}f_{21} + e_{23}f_{31} + e_{24}f_{41} & e_{21}f_{12} + e_{22}f_{22} + e_{23}f_{32} + e_{24}f_{42}\end{bmatrix} \]