In this lecture, we took a short preliminary quiz, and went over some of the motivation behind linear algebra.
The quiz lasted approximately 20 minutes. There were 6 true/false questions on basic linear algebra material, which are summarised below:
- Given a 2x3 matrix A and a 2x2 matrix B, AB is defined but BA is not. Answer: False; BA is defined but AB is not.
- A matrix A is invertible if and only if it has a non-zero determinant. Answer: True.
- This particular matrix (3x3, first column is all 0's) is invertible. Answer: False, because of the first column of all 0's (its determinant is clearly 0).
- A 2x3 matrix is a linear transformation from $\mathbb R^3$ to $\mathbb R^2$. Answer: True. (Multiply the matrix by the transpose of the vector, on the right.)
- 4 vectors in $\mathbb R^3$ form a basis for it. Answer: False; they are clearly not linearly independent.
- Any non-zero polynomial with real coefficients (for example, $x^2+1$) has a real root. Answer: False.
Then, there were questions about what math courses you've taken, what your interest in the course is, and if you have any additional comments.
2Motivation and goals¶
Linear algebra is the foundation for many topics: abstract algebra (which arises naturally from relaxing the definition of a vector space), ODEs, function analysis. It arose out of the need to solve the linear system $A\vec x = \vec b$ which, though simple, has applications in many different fields. Our goal in this course is to study vector spaces as well as the linear maps (or operations, or transformations) between vector spaces.