We focus on solving systems of linear equations. Typically, we have $n$ equations and $n$ unknowns. Two visualizations are key: the Row picture and the Column picture.

• ★The Row picture shows one equation at a time with intersecting lines.0:10
• ★The Column picture visualizes the rows and columns of a matrix.0:15
• ★Matrix representation is denoted as $A$.0:20

Example: Consider $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$ and $\vec{b}=\left(\begin{array}{c}5\\ 6\end{array}\right)$.0:30

We have two equations: $2x-y=0$ and $-x+2y=3$. We can represent these equations in matrix form.

• ★A matrix is a rectangular array of numbers.2:14
• ★The coefficient matrix is $\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$.2:14
• ★The matrix form is $A\vec{x}=\vec{b}$.2:14

Example: For the equations $2x-y=0$ and $-x+2y=3$, we can write them as $A\vec{x}=\vec{b}$.2:14

The Row picture shows points on the xy-plane that satisfy the linear equation $2x-y=0$. The equation represents a straight line.

• ★The equation $2x-y=0$ defines a straight line.4:21
• ★The origin $(0,0)$ is a solution to $2x-y=0$.4:21
• ★Choosing $x=1$ gives the point $(1,2)$ as a solution.4:21
• ★Linear equations have solutions that form a straight line.4:21

Example: For $x=0.5$, we find $y=1$, giving the point $(0.5,1)$ as another solution.4:21

To determine if a line passes through the origin, check if $y=0$. For example, when $y=0$, $x=-3$. Substituting $x=-1$ gives $y=1$. These points help visualize the line.

• ★A line passes through the origin if $y=0$ leads to $x=0$6:23
• ★Substituting values for $x$ yields corresponding $y$ values6:30
• ★The point $(1,2)$ satisfies both equations6:40

Example: For $x=-1$, substituting gives $y=1$, so the point $(-1,1)$ lies on the line.6:35

The row picture shows the solution as the intersection of two lines for two equations and two unknowns. The column picture focuses on the columns of the matrix and how they relate to the equations.

• ★The row picture represents solutions as intersections of lines.8:25
• ★The column picture emphasizes matrix columns and their relationships.8:30
• ★A linear combination involves multiplying columns by coefficients and adding them.8:35
• ★Adjusting coefficients aims to produce a specific vector, like $\left(\begin{array}{c}0\\ 3\end{array}\right)$.8:40

Example: To find a specific vector, adjust coefficients in the linear combination of matrix columns.8:45

Vectors can be visualized in two-dimensional space using their components. For example, the vector $(2,-1)$ moves right 2 and down 1, while the vector $(-1,2)$ moves left 1 and up 2.

• ★A linear combination of vectors combines them to form a new vector.10:31
• ★The vector $(0,3)$ can be formed by $1$ of $(2,-1)$ and $2$ of $(-1,2)$.10:31
• ★Vectors are added by connecting the tip of one vector to the tail of another.10:31

Example: Combining $1$ of $(2,-1)$ and $2$ of $(-1,2)$ results in $(0,3)$.10:31

We perform linear combinations of vectors to find specific coordinates. For example, adding the vector \begin{pmatrix} 1 \ 2 \\ \text{(move left one, up two)} \\ \text{to another vector} \\ \text{results in a new position.} \\ \text{The coordinates are (0, 3).}

• ★Vector addition combines components: $2+(-2)=0$ and $-1+4=3$12:41
• ★Resulting coordinates from addition are $(0,3)$, corresponding to point b.12:41
• ★Visual representation aids understanding of vector combinations.12:41
• ★Exploring all combinations of vectors is key in linear equations.12:41

Example: Adding the vectors $\left(\begin{array}{c}12\end{array}\right)$ and $\left(\begin{array}{c}-14\end{array}\right)$ results in $\left(\begin{array}{c}03\end{array}\right)$.12:41

Linear combinations can fill the entire plane, allowing any right-hand side to be achieved. Transitioning to three equations with three unknowns, we introduce variables $x$, $y$, and $z$. Understanding these equations requires both row and column perspectives.

• ★Linear combinations can achieve any right-hand side in the plane.14:45
• ★Transition from two equations to three equations with three unknowns.14:55
• ★Equations can be understood through row and column pictures.15:10
• ★Matrix form of the equations is significant.15:20

We represent a system of three equations with three unknowns using the matrix $A$. The right-hand side is given by the vector $(0,-1,4)$. Visualizing these equations in three-dimensional space helps identify solution points.

• ★The matrix $A$ represents three equations with three unknowns.16:49
• ★The vector $(0,-1,4)$ is the right-hand side of the equations.16:55
• ★The origin $(0,0,0)$ does not satisfy the second equation.17:00
• ★Solution points include $(1,0,0)$ and $(0,-0.5,0)$.17:10

Example: For $z=1$, the point $(0,0,1)$ is also a solution.17:25

A linear equation in three dimensions represents a plane. Each equation in a system corresponds to a distinct plane. The intersection of two planes forms a line, while three non-parallel planes can intersect at a single point.

• ★The graph of a linear equation in three dimensions is a plane.19:00
• ★The intersection of two planes is a line.19:10
• ★Three non-parallel planes can intersect at a single point.19:20
• ★If planes are parallel, they do not intersect.19:30

Three planes can intersect at a single point, which is the solution to the system of equations. Visualizing these planes in a row picture is more complex than visualizing two lines. The column picture is clearer for understanding linear combinations.

• ★Three planes can intersect at a point, representing a solution.21:08
• ★The left side of the equation is a linear combination of three vectors.21:15
• ★The right side is a vector: zero minus one four.21:20

We visualize the solution to a linear equation using vectors. Given three vectors, we can achieve a target vector by selecting one of the existing vectors directly.

• ★The target vector is one of the existing vectors.23:15
• ★The solution is found by choosing the vector with coefficients $x=0$, $y=0$, $z=1$.23:25

Example: Select the third vector $\left(\begin{array}{c}0\\ -1\\ 4\end{array}\right)$ to achieve the target vector.23:35

The point $(0,0,1)$ represents the intersection of three planes defined by linear equations. Changing the right-hand side of these equations alters the solution, which we will explore in the next lecture.

• ★The point $(0,0,1)$ is the intersection of three planes.25:16
• ★Changing the right-hand side can lead to different solutions.25:25
• ★The row picture illustrates new planes meeting at a different point.25:35

Example: Adding the first two columns creates a new right-hand side, leading to a solution involving one of the original planes.25:30

The equation $Ax=b$ raises the question of whether it can be solved for every vector $\mathbf{b}$. This relates to whether linear combinations of the columns of $A$ can fill three-dimensional space.

• ★The equation $Ax=b$ can represent multiple solutions depending on $\mathbf{b}$.27:24
• ★Linear combinations of columns determine the span in three-dimensional space.27:24

Multiplying a matrix by a vector gives linear combinations of the matrix's columns. We aim to express a vector $\mathbf{b}$ as $A\mathbf{x}=\mathbf{b}$, where $A$ is the matrix and $\mathbf{x}$ is the vector of coefficients.

• ★Multiplying a matrix by a vector results in linear combinations of columns.29:25
• ★A non-singular matrix is invertible.29:35
• ★If columns are coplanar, combinations lie in the same plane.29:45
• ★If elimination fails, the matrix's columns have issues.29:55

A singular matrix does not have an inverse, which means not every right-hand side $b$ has a solution. Only those $b$ that lie in the same plane as the columns of the matrix can be reached.

• ★A singular matrix leads to no solution for every right-hand side $b$.31:36
• ★Only right-hand sides $b$ in the column plane can be solved.31:45
• ★Most right-hand sides are unreachable if outside the column plane.31:55
• ★Random matrices in software like MatLab are often non-singular.32:05

Columns in a matrix must be linearly independent for the matrix to be invertible. If columns are dependent, they do not provide new information, limiting the solutions to linear equations.

• ★A non-singular matrix is invertible, allowing for unique solutions to $Ax=b$.33:45
• ★Dependent columns limit the dimensionality of the span.33:55
• ★Nine independent vectors can fill nine-dimensional space.34:05
• ★If one column is a duplicate, the span is reduced to an eight-dimensional plane.34:15

To multiply a matrix by a vector, we can use two methods: column-wise and row-wise. For a matrix $A$ and vector $x$, we compute $Ax=b$. The column-wise method uses each column of $A$ with the vector $x$.

• ★Matrix multiplication is $Ax=b$.35:49
• ★Column-wise multiplication uses each column of $A$.35:55
• ★Row-wise multiplication involves the dot product.36:05

Example: Using $A$ and $x$, the result of the multiplication is $Ax=\left(\begin{array}{c}12\\ 7\end{array}\right)$.36:15

Matrix multiplication can be interpreted as a dot product and as linear combinations of the columns of the matrix. For a matrix $A$ and a vector $\text{x}$, the product $A\text{x}$ can be visualized using both rows and columns.

• ★The dot product is the sum of products of corresponding elements: $a_1{b}_1+a_2{b}_2$37:55
• ★Matrix multiplication $A\text{x}$ gives a linear combination of the columns of $A$37:58
• ★For the system $A\text{x}=\text{b}$, if $\text{b}=[12,7]$, then $\text{x}=[1,2]$38:01

Example: For the system $A\text{x}=[12,7]$, the solution is $\text{x}=[1,2]$.38:05