向量空间中的正交

函数正交

矩阵/变换正交

其他领域的正交

X 轴

其他正交领域

Y 轴

向量 v

向量 w

θ ≈ 160.9°

剩余角度

夹角θ

7 - 6i

7 - 8i

乘积: (7 - 6i)(7 - 8i)

−3 + 6i

最终乘积: (7 - 6i)(7 - 8i)(−3 + 6i)

复数连线

乘以 (−3 + 6i) 线

(7 - 6i)(7 - 8i)(-3 + 6i) 复数乘法过程

实部

虚部

\u5F53\u524D\u5165\u53E3\u6570\u503C

\u6709\u7406\u540E\u7ED3\u679C

\u7EBF\u6761

Complex Number Rationalization Visualization

Result a) (3/2 - 1/2 i)

Result b) (5/13 + 1/13 i)

Complex Number Division and Simplification Visualization

Real Axis

Imaginary Axis

(3 - i)/2

(5 + i)/13

3/2 - 1/2 i

5/13 + 1/13 i

Real

Imaginary

Expression b) Original

Expression b) After factorization

Expression b) Final result

Expression c) Original numerator

Expression c) Original denominator

Expression c) Final result

Expression c) Vector numerator × conjugate denominator

Expression c) Vector denominator to conjugate

Unit Circle

夹角 θ ≈ 1.305 rad
(74.8°)

夹角 θ

Complex Number (5+3i)

Vector Projection on Real Axis

Vector Projection on Imaginary Axis

Point (5,0)

Point (0,3)

Circle (Magnitude = √34 ≈ 5.831)

函数图像: z<0 时不等式反转

典型点

真值表

真值表标注

x, y

命题变量与真值

z = (x or y) × z

真值

x < y

(-z)x < (-z)y

z x > z y

Inequality Direction Reversal When Multiplying by a Negative Number

Value

Values

P∨Q

P⇒(P∨Q)

命题组合

真值 (0=假, 1=真)

Vector v

Vector w

Vector v + w

Projection of v onto w

向量 v 分量点

向量 w 分量点

角度

补角

X轴

Y轴

\(y=\cos\theta\)

Solution \(\theta\)

Horizontal Line \(y = -\frac{3}{\sqrt{11}}\)

Visualization of \( \theta = \arccos\left(-\frac{3}{\sqrt{11}}\right) \)

\(\theta\) (radians)

\(\cos\theta\)

y = cos(θ)

y = -3/√11

θ = arccos(-3/√11)

θ (radians)

cos(θ)

Matrix C (3x2)

Matrix B (2x1)

Dimension Match

Matrix Multiplication Dimensions: Product C B Defined

矩阵C (3×2)

矩阵B (2×1)

乘积CB (3×1)

匹配维度连接线

Row 1

Row 2

Row 3

Columns

原矩阵 M

最简行阶梯形 RREF(M)

矩阵 M 的化简及最简行阶梯形 (RREF)

增广矩阵元素值

系数与常数

方程组

x = (z+7)/4

x = (z+9)/4

Linear Equation System Analysis: Contradiction Demonstration

Eq1: y - 4x + 2z

Eq2: y + z

Eq3: 3y - 4x + 4z

Verification of Solution in System of Equations

Equation 1

Equation 2

Equation 3 LHS

Equation 3 RHS

Linear Equation Verification and Residual Illustration

变量x

变量y

变量z

近似解点

正确解点

高斯消元法求解线性方程组步骤示意

变量值

变量

Pivot 1

Matrix Cells

Pivot 1 (Col 2, Row 1)

Pivot 2 (Col 3, Row 2)

Zero Row (Row 3)

Reduced Row Echelon Form (RREF) Matrix Visualization

Rows

方程1平面

方程2平面

通解曲线

参数限制示意

x₁

x₂

P→Q

(P∨Q)⇔Q

(P∧Q)→P

集合关系 A⊆B

集合交集 A∩B

集合并集 A∪B

补集关系

三个命题等价证明与集合等价关系示意

Q / 集合成员关系

Set B

Set A

Venn Diagram Circles

Venn Diagram Circles B

Line y=x (Intersection Illustration)

Equivalence of Set Relations: A⊆B, A∩B=A, and A∪B=B

Logical Equivalence

Set Subset (A⊆B)

Set Complement

Logical Equivalences and Set Relationships Visualization

X axis

Y axis

Annotations

Set Theory Equivalence: A⊆B ⇔ A∩B = A ⇔ A∪B = B

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