91Ó°ÊÓ

The Column Space of a matrix, often referred to as the range, is the set of all linear combinations of its column vectors. When we look at a matrix, each column can be seen as a vector under vector space theory. For the matrix we have:\[A = \begin{bmatrix}1 & 2 \3 & 4\end{bmatrix}\]The column space is essentially the "reach" or "span" of the two columns \(\begin{bmatrix}1 \3\end{bmatrix}\) and \(\begin{bmatrix}2 \4\end{bmatrix}\). This span includes all possible vectors \(\mathbf{v}\) that can be produced by a linear combination of these columns, meaning \(\mathbf{v} = c_1\begin{bmatrix}1 \3\end{bmatrix} + c_2\begin{bmatrix}2 \4\end{bmatrix}\) for any scalars \(c_1\) and \(c_2\).
To determine if a vector like \(\mathbf{b}=\begin{bmatrix}5 \6\end{bmatrix}\) is in the column space, you need to find appropriate \(c_1\) and \(c_2\) that satisfy the equation above.

• All linear combinations expressible by the columns
• Represents all space that can be "spanned" by the columns

A Linear Combination in linear algebra is an expression constructed from a set of terms by multiplying each term by a constant and adding the results. The terms can be vectors, matrices, or numbers. In the context of our problem, we are primarily concerned with vectors.
Consider the vector equation:\[\mathbf{b} = c_1 \begin{bmatrix} 1 \3 \end{bmatrix} + c_2 \begin{bmatrix} 2 \4 \end{bmatrix} = \begin{bmatrix} 5 \6 \end{bmatrix}\]Here, the objective is to find constants \(c_1\) and \(c_2\) such that this linear combination equals our vector \(\mathbf{b}\). Finding these constants determines whether the vector \(\mathbf{b}\) lies in the column space of the matrix \(A\).
A vector can be expressed as a linear combination if it lies within the span of the given vectors.

• Involves scalar multiplication and addition
• Expresses dependence among vectors

An Augmented Matrix is a representation of a system of linear equations, combining both the coefficients of the variables and the constants from the equations into a single matrix. In linear algebra, this matrix helps solve problems involving vector equations efficiently.
For our problem:Given the system of equations:\[\begin{align*}1x_1 + 2x_2 &= 5 \3x_1 + 4x_2 &= 6 \\end{align*}\]We transform this into the augmented matrix:\[\begin{bmatrix} 1 & 2 & | & 5 \3 & 4 & | & 6 \end{bmatrix}\]The line separating the coefficients from the constants visually represents the equality of the linear equations, making it easy to apply further operations like row reduction.

• Combines coefficients and constants into one matrix
• Simplifies solving simultaneous equations

Row Reduction, also known as Gaussian elimination, is a method used to solve systems of linear equations. It involves performing elementary row operations on the augmented matrix to achieve a row-echelon form or reduced row-echelon form. The ultimate goal is to simplify the system and find solutions for the variables.For our augmented matrix:\[\begin{bmatrix} 1 & 2 & | & 5 \3 & 4 & | & 6 \end{bmatrix}\]We apply row reduction:

• First, eliminate the \(x_1\) coefficient in the second row by making it zero.
• Next, adjust the matrix to make pivots, with zeros below them, easier to identify.
• Finally, we adjust rows to isolate variables and find their values.

The resulting row-reduced matrix:\[\begin{bmatrix} 1 & 0 & | & -4 \0 & 1 & | & 4.5 \end{bmatrix} \]Indicates solutions \(x_1 = -4\) and \(x_2 = 4.5\), confirming \(b\) is a linear combination of the columns in \(A\).
Row reduction enables analysts to solve linear systems and validate the relationship between a vector and the span of matrix columns.