In this post, my lecture note for Linear Albebra 1, 2 and midterm problems are provided.

Lecture Notes

📄 Download PDF

This lecture note is includes chapter 1 ~ 6 of *Linear Algebra_Friedberg_5th edition*.

This lecture note would be kept updated until it fully covers the book (~ chap 7).

Midterm

Below is the midterm exam for Linear Algebra 2 class which covers chapter 6.

1. For vectorspace $V=R^3$, consist the orthonormal basis for the subspace $W = [(x, y, z) \vert x + 2y + 3z = 0]$.

2. For any orthogonal matrix $A \in M_{3 \times 3}(R)$, show that there exists orthogonal matrix $P$ such that

   \[PAP^{-1} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & cos(\theta) & -sin(\theta) \\ 0 & sin(\theta) & cos(\theta) \\ \end{pmatrix}\]

3. $T$ is a rigid motion on $R^n$, which means, $\lVert T(x)-T(y)\rVert = \lVert x-y\rVert$ for $\forall x, y \in R^n$ $T(0)=0$.

​ (1) Show that $<T(v), T(w)> = <v, w>$. ​ (2) Show that $T$ is a linear operator.

1. Let $V$ be an inner product space, and let $T$ be a normal operator on $V$. Prove that If $\lambda_{1}, \lambda_{2}$ are distince eigenvector of $T$, then their eigenspaces are orthogonal.

2. Solve the questions below.

   (1) For the following matrix $A$, find an orthogonal matrix $P$ and a diagonal matrix $D$ such that $P*AP=D$ \(\begin{pmatrix} 2 & -2 \\ -2 & 5 \\ \end{pmatrix}\)

   (2) Explain whether (0, 0) is an extremal points for $f(x,y)=2x^2 -4xy+ 5y^2$.

3. Find a singular value decomposition for the matrix below. \(\begin{pmatrix} 1 & 0 & -1 \\ 2 & 0 & -2 \\ \end{pmatrix}\)