Common Basis

In mathematics, a common basis refers to a set of vectors that spans the same vector space. A vector space is a collection of vectors that satisfy certain properties, such as closure under addition and scalar multiplication. A basis for a vector space is a set of linearly independent vectors that span the entire space. When two vector spaces share the same basis, they are said to have a common basis.

Properties of a Common Basis

When two vector spaces have a common basis, the following properties hold true:

The common basis spans both vector spaces: Every vector in each vector space can be expressed as a linear combination of the vectors in the common basis.
The common basis is linearly independent in both vector spaces: No vector in the common basis can be expressed as a linear combination of the other vectors in the basis.
The common basis has the same number of vectors in both vector spaces: The dimension of the vector spaces is the same, as the common basis spans both spaces.

Example

Consider two vector spaces V and W with a common basis B. If v is a vector in V and w is a vector in W, then both v and w can be expressed as linear combinations of the vectors in B:

v = a₁b₁ + a₂b₂ + ... + a_nb_n

w = c₁b₁ + c₂b₂ + ... + c_nb_n

where a₁, a₂, ..., a_n and c₁, c₂, ..., c_n are scalars and b₁, b₂, ..., b_n are the vectors in the common basis B.

Study Guide

To understand the concept of a common basis, it's important to grasp the following key points:

Review the definition and properties of a vector space, including closure under addition and scalar multiplication.
Understand the definition of a basis for a vector space and the concept of linear independence.
Study how to express vectors as linear combinations of basis vectors.
Practice identifying common bases for different pairs of vector spaces.
Work through examples of finding common bases and understanding their properties.

By mastering these concepts, you can develop a solid understanding of common bases and their significance in vector spaces.

◂Math Worksheets and Study Guides Eighth Grade. Displaying data

Organizing & Displaying Data

The resources above cover the following skills:

Data Analysis and Probability (NCTM)

Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.

Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

Select and use appropriate statistical methods to analyze data.

Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.

Develop and evaluate inferences and predictions that are based on data.

Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.

Grade 8 Curriculum Focal Points (NCTM)

Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets

Students use descriptive statistics, including mean, median, and range, to summarize and compare data sets, and they organize and display data to pose and answer questions. They compare the information provided by the mean and the median and investigate the different effects that changes in data values have on these measures of center. They understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center. Students select the mean or the median as the appropriate measure of center for a given purpose.

Connections to the Grade 8 Focal Points (NCTM)

Data Analysis: Building on their work in previous grades to organize and display data to pose and answer questions, students now see numerical data as an aggregate, which they can often summarize with one or several numbers. In addition to the median, students determine the 25th and 75th percentiles (1st and 3rd quartiles) to obtain information about the spread of data. They may use box-and-whisker plots to convey this information. Students make scatterplots to display bivariate data, and they informally estimate lines of best fit to make and test conjectures.