docs: update blog 7.29

Author: lijing-22

pages/blog/essential-guide-column-vectors.mdx

@@ -20,7 +20,13 @@ Column vectors are foundational elements in linear algebra, playing a crucial ro
 
 ### What is a Column Vector?
 
-A column vector, denoted as \(\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\), consists of elements arranged in a vertical format. The elements can be integers, real numbers, or complex numbers. The primary distinction between column vectors and other vector types, such as row vectors, lies in their orientation: column vectors are vertical, while row vectors are horizontal.
+A **column vector**, denoted as:
+
+x1, x2, ... xn.
+
+It consists of elements arranged in a vertical format. The elements can be integers, real numbers, or complex numbers.
+
+The primary distinction between column vectors and other vector types, such as **row vectors**, lies in their orientation: column vectors are vertical, while row vectors are horizontal.
 
 ### Defining Properties and Characteristics of Column Vectors
 
@@ -32,7 +38,7 @@ Column vectors possess several key characteristics:
 
 ### Difference Between Column Vectors and Row Vectors
 
-The fundamental difference between column vectors and row vectors lies in their arrangement. A row vector is expressed as \(\begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix}\), whereas a column vector is structured vertically. This distinction impacts various mathematical operations, such as dot product and matrix multiplication, which are defined differently for row and column vectors.
+The fundamental difference between column vectors and row vectors lies in their arrangement. This distinction impacts various mathematical operations, such as dot product and matrix multiplication, which are defined differently for row and column vectors.
 
 ## Mathematical Operations with Column Vectors
 

pages/blog/essential-vectors-data-field.mdx

@@ -26,17 +26,7 @@ A vector is a mathematical object characterized by both magnitude and direction.
 - **Row Vectors**: A matrix with a single row.
 - **Zero Vectors**: A vector with all components equal to zero.
 
-The mathematical properties of vectors, such as addition, subtraction, and scalar multiplication, are foundational for numerous data manipulation techniques. For instance, if we have two vectors \( \mathbf{A} \) and \( \mathbf{B} \):
-
-\[
-\mathbf{A} = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}
-\]
-
-The sum of vectors \( \mathbf{A} \) and \( \mathbf{B} \) is calculated as follows:
-
-\[
-\mathbf{A} + \mathbf{B} = \begin{bmatrix} a_1 + b_1 \\ a_2 + b_2 \\ a_3 + b_3 \end{bmatrix}
-\]
+The mathematical properties of vectors, such as addition, subtraction, and scalar multiplication, are foundational for numerous data manipulation techniques.
 
 ### Importance of Vectors in Data Analysis
 
@@ -51,11 +41,7 @@ In machine learning, vectors serve as the backbone for feature representation an
 
 #### Feature Representation and Model Input
 
-When training machine learning models, data must be transformed into a numerical format. This transformation typically involves representing features as vectors. For example, consider a dataset of houses with features like square footage, number of bedrooms, and age. Each house can be represented as a vector:
-
-\[
-\mathbf{X} = \begin{bmatrix} \text{Square Footage} \\ \text{Bedrooms} \\ \text{Age} \end{bmatrix}
-\]
+When training machine learning models, data must be transformed into a numerical format. This transformation typically involves representing features as vectors. For example, consider a dataset of houses with features like square footage, number of bedrooms, and age.
 
 By representing features as vectors, machine learning algorithms can efficiently process and analyze the data.
 
@@ -128,13 +114,7 @@ Predictive analytics relies heavily on vectors to build and analyze models. Let'
 
 #### Building Predictive Models with Vectors
 
-In predictive modeling, vectors represent input features, while the output is often a scalar or another vector. For instance, in a linear regression model, the prediction can be expressed as:
-
-\[
-\mathbf{y} = \mathbf{X} \cdot \mathbf{w} + b
-\]
-
-Where \( \mathbf{y} \) is the output vector, \( \mathbf{X} \) is the feature vector, \( \mathbf{w} \) represents weights, and \( b \) is the bias.
+In predictive modeling, vectors represent input features, while the output is often a scalar or another vector.
 
 #### Vectors in Time Series Analysis
 

pages/blog/how-to-remove-table-formatting-in-excel.mdx

@@ -1,7 +1,7 @@
 ---
 title: "How to Effectively Remove Table Formatting in Excel"
 description: "Discover how to remove table formatting in Excel with our step-by-step guide, simplifying your data manipulation and enhancing performance. Learn effective manual and automated methods, including using VBA and Chat2DB, to streamline your data management tasks effortlessly."
-image: ""
+image: "https://i.ibb.co/Psdcd4ZC/9b47eb705337.jpg"
 category: "Guide"
 date: July 29, 2025
 ---