---
Created: 2025-12-09
Type: Zettel
aliases:
References:
Links:
- "[[Practice Midterm 1 Work]]"
- "[[Practice Midterm 2 Work]]"
- "[[Practice Final Work]]"
tags:
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## Past exams
### Midterm 1
- Problem 1
- Intersection of sub spaces is also a subspace
- Set that spans $n$ dimensions will have at least $n$ linearly independent elements
- A linear map can only have multiple inputs map to the same output if and only if they map to zero. There is no other possible overlap of outputs. Therefore, T for some $T(x)\neq 0$ is injective.
- Problem 2
- Missed non empty in the definition of linear subspace
- Mistake in proving that $S_{1}+S_{2}$ is a linear subspace of $\mathbb{R}^n$
- Problem 3
- Mistake in solving system of linear equations using matrix
- Mistake in computing inverse of matrix
- Problem 4
- Finding matrix representation of transformation from example calculations
- Definition of linear map
- Problem 5
- Mistake in defining make from $\mathbb{N}^3\to \mathbb{N}$ for Schroder-Bernstein
- Did not define a map from $\mathbb{N}\to \mathbb{N}^3$
### Midterm 2
- Problem 1
- Definition of $S^\perp$
- Finding $S^\perp$ for some set
- Showing linearity of transformation that uses the dot product (bi linearity)
- Problem 2
- Definition of cross product in $\mathbb{R}^3$ with determinant
- Showing orthogonality of cross product
- Calculating 4 by 4 determinant
- Problem 3
- Definition of characteristic polynomial
- Definition of eigenvalues and eigenvectors, $\lambda \in \mathbb{C}$, $v \neq \vec{0}$
- Compute eigenvalues for matrix, and find eigenvalues for matrix
## Practice midterms and finals
### Practice midterm 1
- Relations, equivalence relations
- Binary relation on $T:\mathbb{R}^n\to \mathbb{R}^m$ if and only if $v-w\in \ker T$
- Checking is certain sets are subsets or not
- Definition of a coordinate with respect to a basis
- Properties of linear transformations, kernel, and rank-nullity theorem
- Finding matrix representation of rotation, inclusion, and projection linear maps
### Practice midterm 2
- Determinant of 4 by 4 matrix
- Prove that similar matrices have the same characteristic polynomial and the same eigenvalues
### Practice Final
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## Cheat sheet content
- Definition and example of [[Countable & uncountable sets#Power sets|power sets]]
- Definition of Equivalence [[Relation]]
- Definition of a linear subspace
- Definition of a linear transformation
- Properties of a linear transformation
- Schroder-Bernstein theorem, examples for $\mathbb{N}^{2}$, $\mathbb{N}^3$.
- Example of Cantor's diagonal trick
- Definition of rank-nullity
- Definition of cross product
- Definition of dot product in $\mathbb{R}^n$
- Definition of eigenvalues and eigenvectors
- Equations relating to eigenvalues and eigenvectors
- Definition of similar matrices
- Proof of how similar matrices have the same characteristic polynomial and the same eigenvalues
- Definition of the orthogonal complement $S^\perp$
- Sketch of proof showing that $S^\perp$ is a linear subspace
- Equation for Gram-Schmidt process
- Proof that for a an orthonormal matrix $A$, that $A^TA=I$
- Sketch of proof showing that $L(S) \cap S^\perp=0$ and that $L(S)+S=\mathbb{R}^n$
- Equation and example of diagonalization
- Conditions for a matrix to be diagonalizable
- Properties of symmetric matrices
- Answers to the final review
- Anti-symmetric matrices
- Eigenvalue real and complex multiplicity, simple spectrum, symmetric matrices have simple spectrum
- Modified eigenvalues for squared matrices, inverse matrices
- Properties of stochastic matrices
- Will have an eigenvalue 1 among others
- Maximum value for the eigenvalue will be 1, and multiplicity of 1
- General proof writing tips
- Picking something in a set and using general properties of items in a set to prove something
- To show that two sets are equal, show that they are subsets of each other. Pick an element in generality from one set and show that it always belongs in the other.
- Proof by induction: show a base case, then a $n$ case, and then a $n+1$ case
- Proof by contradiction
