91Ó°ÊÓ

Vector spaces are fundamental structures in linear algebra. They consist of a collection of vectors, which can be added together and multiplied by scalars (real numbers), following certain rules. For a set to qualify as a vector space, it must satisfy ten specific properties, including associativity and commutativity of addition, the existence of an additive identity, and distributive properties between scalars and vectors.

In the context of this exercise, the vectors provided, \( w_1, w_2, w_3, \) and \( w_4 \), form a part of the vector space. We are tasked with determining whether another vector, \( b \), can be expressed within this vector space through a specific combination of these vectors.

A linear combination refers to the process of multiplying each vector by a scalar and then summing the results. In mathematical terms, if you have vectors \( w_1, w_2, \ldots, w_n \), any vector \( b \) that can be formed by \( c_1w_1 + c_2w_2 + \ldots + c_nw_n \) is a linear combination of these vectors.

This exercise asks us to decide if a vector \( b \) is expressible as a linear combination of the given vectors \( w_1, w_2, \ldots, w_n \). This involves solving the equation \( Ax = b \), where \( A \) is the matrix whose columns are \( w_1, w_2, \ldots, w_n \). Solving this equation helps determine if such scalars \( c_1, c_2, \ldots, c_n \) exist to express \( b \).

Row reduction, also known as Gaussian elimination, is a key method in linear algebra used to simplify matrices to determine solutions to linear equations. This method involves performing operations on the rows of a matrix to bring it into a simpler form. Specifically, a matrix is reduced to row echelon form or even further simplified to reduced row echelon form.

In the context of this problem, row reduction begins with setting up an augmented matrix that combines matrix \( A \) and vector \( b \). Then, through a series of operations like swapping rows, multiplying by scalars, and adding/subtracting rows, we simplify the matrix to solve for the unknown vector \( x \). Through analysis of the resulting matrix, you can determine if the system is consistent and if \( b \) is indeed a linear combination of the \( w \) vectors.

An augmented matrix is an essential tool in solving systems of linear equations. It's constructed by appending the column vector representing the constant terms (in this case, \( b \)) to the coefficient matrix \( A \). For example, if \( A \) is a 3x3 matrix and \( b \) is a vector of dimension 3, then the augmented matrix will be 3x4.

Using the augmented matrix, we apply row reduction methods to find solutions for \( x \) in the linear equation \( Ax = b \). In problem (a), the matrix \( A \) augmented by \( b \) forms a straight path forward to determine if a solution exists. A consistent system—a row in the form of \( 0x_1 + 0x_2 + 0x_3 = 0 \) in row-reduced form—indicates \( b \) is part of the vector space spanned by \( w_1, w_2, \) and \( w_3 \). For problem (b), the presence of the zero vector in \( A \) indicates potential issues in forming a general solution, often resulting in inconsistencies regardless of \( b \).