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• [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] 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