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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