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[Finite-Dimensional Linear Algebra] 4.1 + 4.2 + 4.3 The Determinant Function Analysis of matrices is central in the study of finite-dimensional linear algebra, because ANY linear operator mapping one finite-dimensional space into another CAN be represented by a matrix. The simplest kind of matrix ia a diagonal matrix. The matrix `A \in F^{m \times n}` is diagonal if `A_{ij} = 0` for all `i \neq j` To simplify matrix-based calculations, we want to transform matrices into .. Show More
[Finite-Dimensional Linear Algebra] 2.5 + 2.6 + 2.7 Basis and Dimension Lemma For a set of vectors ` u_1, u_2,\dots , u_n` in vector space `V`, if `v \in sp \{ u_1, u_2,\dots u_n \} `, then`` sp \{ u_1, u_2,\dots u_n, v \} = sp \{ u_1, u_2,\dots u_n \} ``This leads to the concept of a basis, a spanning set containing the fewest possible elements. Definition of a BasisLet `u_1, u_2,\dots u_n` be vectors in a vector space `V`. We say that `\{ u_1, u_2,\dots u_n \}` is.. Show More
[MIPS] MIPS Registers & Instructions RegistersNameNumberDescription$zero, $00contains the value 0$at1reserved for assembler$v0 - $v12-3values returned by functions$a0 - $a34-7arguments to functions$t0 -  $t78-15temporary variables $s0 - $s716-23saved values $t8 - $t924-25more temporary registers$sp29stack pointer to the top of stack$ra31return addressTemporary vs. Saved RegistersMIPS convention specifies how the registers are suppo.. Show More
[Finite-Dimensional Linear Algebra] 2.4 Linear Combinations and Spanning Sets Definition of a Linear CombinationLet `V` be a vector space over a field `F`, let `u_1, u_2,\dots u_k` be vectors in `V`, and let `\alpha _1, \alpha _2,\dots \alpha _k` be scalars in `F`. Then the linear combination of the vectors with scalars we call weights is: ``\alpha _1 u_1+\alpha _2u_2+\cdots + \alpha _ku_k``Any linear combination of vectors in subspace `S` belongs to `S`, as subspaces are.. Show More
[Finite-Dimensional Linear Algebra] 2.2 Vector Spaces + 2.3 Subspaces The elements of vector space `V` are called vectors, and the elements of the corresponding field `F` are called scalars. Definition of a Vector SpaceLet `F` be a field and let `V` be a nonempty set with two operations defined with respect to these sets: (vector) addition: `u, v \in V \Rightarrow  u + v \in V`scalar multiplication: `\alpha \in F, v \in V \Rightarrow \alpha v \in V` `V` is a vecto.. Show More
[Finite-Dimensional Linear Algebra] 2.1 Fields Definition of a fieldLet `F` be a nonempty set with two defined operations for `\alpha , \beta \in F` called addition and multiplication.addition: `\alpha + \beta \in F`multiplication: `\alpha\beta \in F` `F` is a field iff (if and only if) the operations satisfy the following properties: A. Axioms for AdditionAssociative property of additionfor all `\alpha , \beta , \gamma \in F`, ``(\alpha + \.. Show More
[MIPS] Recursion Example - Fibonacci In MIPS, we can implement recursive functions using stack. The stack is necessary to keep track of values in between calls. Let's try implementing the Fibonacci function. Pseudocode in C:int fib (int n) { if (n Code in MIPS Assembly: We will learn how to write a recursive function in MIPS assembly language using two approaches: an intuitive approach and the standard approach. For the intuitive a.. Show More
[MIPS] Stacks In MIPS architecture, the stack grows downward in terms of memory -- when an element is pushed into a stack, it should be stored in an address lower than that of the previous element. We assume the address of the topmost (newest) element in the stack is the one stored in the stack pointer register $sp. There are several ways to implement the push or pop instructions. Below are some examples. Pus.. Show More

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