Midterm Review: Systems, Matrices, Vectors, Spaces, and Transformations

students, this lesson brings together the big ideas from your Linear Algebra midterm review: systems of equations, matrices, vectors, vector spaces, and linear transformations 📘✨. These topics may have seemed separate at first, but they are deeply connected. A system of equations can be rewritten using matrices. Matrices can act on vectors. Vectors can live inside spaces. And linear transformations describe how one vector space changes into another.

What You Need to Know for the Midterm

The main goal of this review is to help you see how all the pieces fit together. By the end, you should be able to explain the core ideas, use standard procedures, and recognize how one topic leads naturally into the next.

You should be able to answer questions like:

How do we represent a system of linear equations using an augmented matrix?
What does it mean for a vector to be in the span of other vectors?
How do matrices represent transformations?
What is the difference between a vector space and a subspace?
Why is every linear transformation tied to a matrix?

For example, if a store sells $x$ sandwiches and $y$ salads, and the total number of items and total cost are known, that situation can become a system of equations. Then you can rewrite it in matrix form and use row reduction to solve it. That same kind of thinking shows up throughout this unit ✅

Systems of Linear Equations and Row Reduction

A system of linear equations is a set of equations involving the same variables. A solution is a set of values that makes every equation true at the same time. In linear algebra, we often rewrite a system in matrix form because matrices make patterns easier to see.

For example, consider the system

$\begin{aligned}$

$2x+y&=5 \\$

$x-y&=1$

$\end{aligned}$

We can write the augmented matrix as

$\begin{bmatrix}$

2 & 1 & 5 \\

1 & -1 & 1

$\end{bmatrix}$

Using row reduction, we can find the solution. The key idea is that row operations do not change the solution set. They only rewrite the system in a simpler form.

A system can have:

one solution, when the lines or planes meet at a single point
no solution, when the equations contradict each other
infinitely many solutions, when the equations describe the same relationship or a family of relationships

A common midterm skill is interpreting the reduced row echelon form, often called RREF. If a row looks like $[0\ 0\ 0\ |\ c]$ where $c\neq 0$, then the system is inconsistent. If there are free variables, then there are infinitely many solutions.

Real-world example: a school fundraiser may track ticket sales and snack sales. If each ticket and snack has a price, the totals can be modeled with equations. Solving the system tells you how many of each item were sold 🎟️🍿

Matrices as Tools for Organizing and Solving

A matrix is a rectangular array of numbers. In linear algebra, matrices are not just tables. They are tools for organizing information and performing operations efficiently.

The entries of a matrix can represent coefficients in a system, coordinates of vectors, or the rule of a transformation. For example, the matrix

$A=\begin{bmatrix}$

1 & 2 \\

3 & 4

$\end{bmatrix}$

can be used in several ways. It can be part of a system, or it can define a transformation that sends vectors in $\mathbb{R}^2$ to new vectors in $\mathbb{R}^2$.

Important matrix ideas for the midterm include:

matrix addition and scalar multiplication
matrix multiplication
the identity matrix $I$
invertible matrices
the meaning of $A^{-1}$ when it exists

Matrix multiplication is especially important. If $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, then $AB$ is an $m\times p$ matrix. The dimensions must match. This matters because matrix multiplication represents composition of transformations and combining systems.

A matrix is invertible if there exists a matrix $A^{-1}$ such that

$AA^{-1}=A^{-1}A=I$

If $A$ is invertible, then the system $A\mathbf{x}=\mathbf{b}$ has a unique solution for every vector $\mathbf{b}$ in the appropriate space. That fact connects matrices to systems and to transformations.

Vectors, Span, and Linear Combinations

A vector is an object with magnitude and direction, or in algebra class, usually an ordered list of numbers. A vector in $\mathbb{R}^n$ looks like

$\mathbf{v}=\begin{bmatrix}v_1\v_2\\vdots\v_n\end{bmatrix}$

Vectors can represent position, velocity, force, or data values. In linear algebra, they are the building blocks of spaces.

A linear combination is an expression of the form

$ c_1\mathbf{v}_1+c_2\mathbf{v}_2+\cdots+c_k\mathbf{v}_k$

where the $c_i$ are scalars. The span of vectors is the set of all linear combinations of those vectors. If a vector lies in the span of others, it means it can be built from them.

Example: if

$\mathbf{u}$=$\begin{bmatrix}1$\\$0\end{bmatrix}$,\quad $\mathbf{v}$=$\begin{bmatrix}0$\\$1\end{bmatrix}$

then any vector in $\mathbb{R}^2$ can be written as

$a\mathbf{u}+b\mathbf{v}$

for some scalars $a$ and $b$. That means $\mathbf{u}$ and $\mathbf{v}$ span $\mathbb{R}^2$.

A key midterm question is whether a given vector is in a span. This is the same as asking whether a system has a solution. For instance, if you want to know whether

$\mathbf{b}$

is in the span of the columns of a matrix $A$, you ask whether $A\mathbf{x}=\mathbf{b}$ is solvable. This connection is one of the most important ideas in the course 🔗

Vector Spaces and Subspaces

A vector space is a set of vectors that is closed under vector addition and scalar multiplication, and it satisfies the usual algebraic rules. Common examples include $\mathbb{R}^n$, the set of all polynomials of degree at most $n$, and the set of all $m\times n$ matrices.

A subspace is a smaller vector space inside a larger one. To check whether a set is a subspace, it must:

contain the zero vector
be closed under addition
be closed under scalar multiplication

For example, the set of all vectors in $\mathbb{R}^3$ of the form

$\begin{bmatrix}$

$a\b\\0$

$\end{bmatrix}$

is a subspace of $\mathbb{R}^3$. It is the $xy$-plane through the origin. But the set of vectors of the form

$\begin{bmatrix}$

$a\b\\1$

$\end{bmatrix}$

is not a subspace, because it does not contain the zero vector.

This matters on the midterm because many concepts are really about subspaces:

the column space is the span of the columns of a matrix
the null space is the set of all solutions to $A\mathbf{x}=\mathbf{0}$
the row space is the span of the rows of a matrix

These subspaces help describe what a matrix can do and what solutions are possible.

Linear Transformations and Their Matrices

A linear transformation is a function $T$ that maps vectors to vectors and preserves addition and scalar multiplication. That means

$T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$

and

$T(c\mathbf{v})=cT(\mathbf{v})$

for any vectors $\mathbf{u}$, $\mathbf{v}$ and scalar $c$.

Common examples include rotations, reflections, projections, and stretches. For instance, a transformation might rotate every vector in the plane by $90^\circ$. Another might compress all vectors onto a line.

Every linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ can be represented by a matrix $A$ such that

$T(\mathbf{x})=A\mathbf{x}$

This is a major bridge between abstract ideas and computation. If you know the matrix, you can compute the output of the transformation. If you know the transformation, you can often build its matrix.

Example: if the standard basis vectors are

$\mathbf{e}_1$=$\begin{bmatrix}1$\\$0\end{bmatrix}$,\quad $\mathbf{e}_2$=$\begin{bmatrix}0$\\$1\end{bmatrix}$

and a transformation sends $\mathbf{e}_1$ to $\begin{bmatrix}2\\1\end{bmatrix}$ and $\mathbf{e}_2$ to $\begin{bmatrix}0\\3\end{bmatrix}$, then the matrix of the transformation is

$A=\begin{bmatrix}$

2 & 0 \\

1 & 3

$\end{bmatrix}$

because the columns are the images of the basis vectors.

How the Big Ideas Connect

The best midterm review strategy is to see the chain:

a system of equations can be written as $A\mathbf{x}=\mathbf{b}$
the columns of $A$ determine whether $\mathbf{b}$ is in the column space of $A$
vectors and their spans describe possible combinations
vector spaces and subspaces organize those combinations
linear transformations describe how vectors move from one space to another
matrices are the computational tool that ties everything together

This is why a question about one topic often requires knowledge of another. For example, if you are asked whether a transformation is one-to-one, you may need to think about the null space. If you are asked whether a vector is reachable by a matrix, you may need to think about span and consistency of a system.

Real-world example: in computer graphics, a matrix can rotate an image, scale it, or shift its shape. The pixels are treated like vectors, and the transformation changes their positions while preserving the rules of linearity. This is linear algebra in action 🎨💻

Conclusion

students, the core idea of this unit is that linear algebra is one connected system. Equations, matrices, vectors, spaces, and transformations are not isolated topics. They describe different views of the same mathematical structure. For the midterm, focus on translating between forms: equation to matrix, matrix to transformation, vector to span, and system to solution set. If you can move confidently between these viewpoints, you are ready to show strong reasoning and apply the ideas to new problems ✅

Study Notes

A system of linear equations can be written as an augmented matrix and solved with row reduction.
Row operations preserve the solution set.
A system can have one solution, no solution, or infinitely many solutions.
A vector in $\mathbb{R}^n$ is an ordered list of numbers.
A linear combination has the form $c_1\mathbf{v}_1+\cdots+c_k\mathbf{v}_k$.
The span of vectors is the set of all their linear combinations.
A vector is in the span of columns of $A$ exactly when $A\mathbf{x}=\mathbf{b}$ has a solution.
A vector space is a set closed under addition and scalar multiplication.
A subspace must contain the zero vector and be closed under addition and scalar multiplication.
The column space, row space, and null space are all important subspaces.
A linear transformation satisfies $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$ and $T(c\mathbf{v})=cT(\mathbf{v})$.
Every linear transformation can be represented by a matrix.
The matrix columns tell how the transformation acts on basis vectors.
These topics are all connected and often appear together on the midterm.