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$\mathbf{y}$:

$$\mathbf{y}=\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]$$

Write down its transpose ${\mathbf{y}}^{\mathrm{\top }}$

Solution:

• The transpose of the vector $\mathbf{y}$ is:

$${\mathbf{y}}^{\mathrm{\top }}=\left[\begin{array}{ccc}{y}_1& y_2& y_3\end{array}\right]$$

Exercise 2

Given the following matrix $\mathbf{N}$:

$$\mathbf{N}=\left[\begin{array}{cc}{n}_{11}& n_{12}\\ n_{21}& n_{22}\\ n_{31}& n_{32}\end{array}\right]$$

Write down its transpose, ${\mathbf{N}}^{\mathrm{\top }}$

Solution:

• The transpose of the matrix $\mathbf{N}$ is:

$${\mathbf{N}}^{\mathrm{\top }}=\left[\begin{array}{ccc}{n}_{11}& n_{21}& n_{31}\\ n_{12}& n_{22}& n_{32}\end{array}\right]$$

Matrix operations

Exercise 3

Consider the following matrices $\mathbf{C}$ and $\mathbf{D}$:

$$\mathbf{C}=\left[\begin{array}{ccc}{c}_{11}& c_{12}& c_{13}\\ c_{21}& c_{22}& c_{23}\end{array}\right],\mathbf{D}=\left[\begin{array}{cc}{d}_{11}& d_{12}\\ d_{21}& d_{22}\\ d_{31}& d_{32}\end{array}\right]$$

• What are the dimensions of $\mathbf{C}$?

• What are the dimensions of $\mathbf{D}$?

• For the matrix product $\mathbf{C}\mathbf{D}$:

  1. Determine if the product is valid, and explain why.

  2. 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 $\mathbf{C}\mathbf{D}$:

  1. 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).

  2. The dimensions of the resulting matrix will be $2\times 2$.

Exercise 4

Consider the following matrices $\mathbf{E}$ and $\mathbf{F}$:

$$\mathbf{E}=\left[\begin{array}{cccc}{e}_{11}& e_{12}& e_{13}& e_{14}\\ e_{21}& e_{22}& e_{23}& e_{24}\end{array}\right],\mathbf{F}=\left[\begin{array}{cc}{f}_{11}& f_{12}\\ f_{21}& f_{22}\\ f_{31}& f_{32}\\ f_{41}& f_{42}\end{array}\right]$$

• What are the dimensions of $\mathbf{E}$?

• What are the dimensions of $\mathbf{F}$?

• For the matrix product $\mathbf{E}\mathbf{F}$:

  1. Determine if the product is valid, and explain why.

  2. 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 $\mathbf{E}\mathbf{F}$:

  1. 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).

  2. The resulting matrix is:

$$\begin{gathered} \mathbf{E}\mathbf{F}=\left[\begin{array}{cccc}{e}_{11}& e_{12}& e_{13}& e_{14}\\ e_{21}& e_{22}& e_{23}& e_{24}\end{array}\right]\left[\begin{array}{cc}{f}_{11}& f_{12}\\ f_{21}& f_{22}\\ f_{31}& f_{32}\\ f_{41}& f_{42}\end{array}\right]= \\ \left[\begin{array}{cc}{e}_{11}{f}_{11}+e_{12}{f}_{21}+e_{13}{f}_{31}+e_{14}{f}_{41}& e_{11}{f}_{12}+e_{12}{f}_{22}+e_{13}{f}_{32}+e_{14}{f}_{42}\\ e_{21}{f}_{11}+e_{22}{f}_{21}+e_{23}{f}_{31}+e_{24}{f}_{41}& e_{21}{f}_{12}+e_{22}{f}_{22}+e_{23}{f}_{32}+e_{24}{f}_{42}\end{array}\right] \end{gathered}$$