Linear Algebra - Complete Course Study Guide Flashcards

A collection of 120 flashcards covering vectors, matrices, systems of equations, determinants, vector spaces, and eigenvalues from a Linear Algebra course study guide.

What is the definition of a vector according to the transcript?

An ordered list of numbers that represents a quantity with magnitude and direction.

What is a scalar?

An ordinary number used to scale a vector.

How are vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$ commonly written?

$\mathbf{v} = \langle v_1, v_2 \rangle$ in $\mathbb{R}^2$ and $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$ in $\mathbb{R}^3$.

How is vector addition performed?

By adding the corresponding components of the vectors: $\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2, \dots \rangle$.

How is scalar multiplication performed on a vector?

By multiplying every component of the vector by the scalar: $c \mathbf{v} = \langle c v_1, c v_2, \dots \rangle$.

When are two vectors considered equal?

When all of their corresponding components are equal.

What characterizes the zero vector?

All of its components are equal to zero.

What is the formula for the magnitude or norm of a vector $\mathbf{v}$?

$$||\mathbf{v}|| = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2}$$

What is the length of a unit vector?

$$1$$

How do you normalize a nonzero vector $\mathbf{v}$?

Divide the vector by its magnitude: $\mathbf{u} = \frac{\mathbf{v}}{||\mathbf{v}||}$.

If $\mathbf{v} = \langle 3, 4 \rangle$, what is its magnitude?

$$||\mathbf{v}|| = 5$$

What is the unit vector in the direction of $\mathbf{v} = \langle 3, 4 \rangle$?

$$\langle \frac{3}{5}, \frac{4}{5} \rangle$$

What condition must two nonzero vectors meet to be considered parallel?

One must be a scalar multiple of the other.

Why are the vectors $\langle 2, 4 \rangle$ and $\langle 1, 2 \rangle$ parallel?

Because $\langle 2, 4 \rangle = 2 \langle 1, 2 \rangle$.

What is the formula for the dot product of two vectors $\mathbf{u}$ and $\mathbf{v}$?

$$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \dots + u_n v_n$$

What is the scalar result of a dot product?

A scalar value.

When are two vectors orthogonal (perpendicular)?

If and only if their dot product is zero ($\mathbf{u} \cdot \mathbf{v} = 0$).

What is the formula for the angle $\theta$ between two nonzero vectors $\mathbf{u}$ and $\mathbf{v}$?

$$\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| ||\mathbf{v}||}$$

What does the projection of $\mathbf{a}$ onto $\mathbf{b}$ ($\text{proj}_b(\mathbf{a})$) measure?

The part of $\mathbf{a}$ that points in the direction of $\mathbf{b}$.

What is the formula for $\text{proj}_b(\mathbf{a})$?

$$\text{proj}_b(\mathbf{a}) = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \right) \mathbf{b}$$

What practical applications are mentioned for vector projection?

Geometry, least-squares problems, mechanics, and decomposing vectors.

If $\mathbf{u} \cdot \mathbf{v} > 0$, what is the nature of the angle between them?

The angle is acute.

If $\mathbf{u} \cdot \mathbf{v} < 0$, what is the nature of the angle between them?

The angle is obtuse.

If $\mathbf{u} \cdot \mathbf{v} = 0$, what is the angle between the vectors?

$90^{\circ}$ (90 degrees).

Given a unit vector $\mathbf{u}$, how do you find the scalar component of $\mathbf{v}$ in the direction of $\mathbf{u}$?

By calculating $\mathbf{v} \cdot \mathbf{u}$.

In which dimension is the cross product $\mathbf{u} \times \mathbf{v}$ defined?

$$\mathbb{R}^3$$

What is the formula for the cross product $\mathbf{u} \times \mathbf{v}$?

$$\mathbf{u} \times \mathbf{v} = \langle u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1 \rangle$$

How is the direction of the cross product vector determined?

By the right-hand rule.

What is the trigonometric formula for the magnitude of the cross product?

$$||\mathbf{u} \times \mathbf{v}|| = ||\mathbf{u}|| ||\mathbf{v}|| \sin(\theta)$$

What geometric area is calculated by $||\mathbf{u} \times \mathbf{v}||$?

The area of the parallelogram formed by $\mathbf{u}$ and $\mathbf{v}$.

How do you find the area of the triangle formed by vectors $\mathbf{u}$ and $\mathbf{v}$?

$$\frac{1}{2} ||\mathbf{u} \times \mathbf{v}||$$

If $\mathbf{u} \times \mathbf{v} = 0$ for nonzero vectors, what can be concluded?

The vectors $\mathbf{u}$ and $\mathbf{v}$ are parallel.

What two items determine a line in 3D space?

A point $\mathbf{r}_0$ and a direction vector $\mathbf{v}$.

What is the vector equation of a line in 3D?

$$\mathbf{r} = \mathbf{r}_0 + t \mathbf{v}$$

What are the parametric equations for a line passing through $\mathbf{r}_0 = \langle x_0, y_0, z_0 \rangle$ with direction $\mathbf{v} = \langle a, b, c \rangle$?

$x = x_0 + at$, $y = y_0 + bt$, $z = z_0 + ct$

How are symmetric equations of a line formed?

By solving each parametric equation for $t$, resulting in $\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$ (where $a, b, c \neq 0$).

What two items determine a plane in 3D?

A point $(x_0, y_0, z_0)$ and a nonzero normal vector $\mathbf{n} = \langle a, b, c \rangle$.

What is the equation of a plane?

$a(x-x_0) + b(y-y_0) + c(z-z_0) = 0$ or $ax + by + cz = d$

When are two planes considered parallel?

When their normal vectors are parallel.

When are two planes considered perpendicular?

When their normal vectors are perpendicular.

What is the condition for a line to be parallel to a plane?

Its direction vector must be orthogonal to the plane's normal vector.

What is the condition for a line to be perpendicular to a plane?

Its direction vector must be parallel to the plane's normal vector.

How is the distance between points $P$ and $Q$ calculated?

$$||Q - P||$$

What is the distance formula from point $(x_0, y_0, z_0)$ to the plane $ax + by + cz = d$?

$$\frac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}}$$

What are skew lines?

Lines in $\mathbb{R}^3$ that are not parallel and do not intersect.

What are the dimensions of an $m \times n$ matrix?

$m$ rows and $n$ columns.

What is a square matrix?

A matrix with the same number of rows and columns.

What is a diagonal matrix?

A square matrix with zeros everywhere except on the main diagonal.

What is a triangular matrix?

A matrix that has zeros either above or below the main diagonal.

What characterizes the identity matrix $I$?

It has $1$s on the diagonal and $0$s elsewhere.

What is the zero matrix?

A matrix where all entries are equal to zero.

What is required to add two matrices?

They must be of the same size.

When does the matrix product $AB$ exist?

When the number of columns of $A$ equals the number of rows of $B$.

Is matrix multiplication commutative in general?

No, in general, $AB \neq BA$.

What are the laws of associativity and distributivity for matrix multiplication?

Associative: $A(BC) = (AB)C$; Distributive: $A(B+C) = AB + AC$.

What is the transpose protocol for entries of a matrix?

$(A^T)_{ij} = A_{ji}$ (rows and columns are switched).

What is the trace of a square matrix ($\text{tr}(A)$)?

The sum of its diagonal entries: $a_{11} + a_{22} + \dots + a_{nn}$.

How is a system of linear equations written compactly?

$$\mathbf{Ax} = \mathbf{b}$$

What does an augmented matrix $[A | b]$ contain?

The coefficient matrix and the constants.

What are the three elementary row operations?

1. Interchange two rows; 2. Multiply a row by a nonzero scalar; 3. Replace a row by itself plus a multiple of another row.

What is a property of row echelon form?

All nonzero rows are above zero rows, and leading entries move to the right as you move down.

What is the specific requirement for Reduced Row Echelon Form (RREF)?

Each pivot must be $1$ and be the only nonzero entry in its column.

In matrix terminology, what is a pivot?

A leading nonzero entry.

What is the difference between a pivot variable and a free variable?

A pivot variable corresponds to a pivot column, while a free variable corresponds to a non-pivot column.

What is usually the result of having free variables in a consistent system?

Infinitely many solutions.

What row reduction result indicates no solution?

A contradiction such as $[0 \, 0 \, 0 \, | \, 5]$.

What is a unique solution condition in terms of variables?

Every variable is a pivot variable and the system is consistent.

What is a homogeneous system?

A system of the form $\mathbf{Ax} = \mathbf{0}$.

Why is a homogeneous system always consistent?

Because $\mathbf{x} = \mathbf{0}$ (the trivial solution) is always a solution.

When does a homogeneous system have a nontrivial solution?

When at least one free variable occurs.

For a square matrix $A$, when does $\mathbf{Ax} = \mathbf{0}$ have only the trivial solution?

Exactly when $A$ is invertible.

What is the definition of the inverse of a square matrix $A$?

A matrix $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$.

What is the determinant condition for matrix invertibility?

A square matrix is invertible if and only if its determinant is nonzero.

What is the inverse formula for a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

$$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$

How do you find $A^{-1}$ using row reduction?

Row reduce the augmented matrix $[A | I]$ to $[I | A^{-1}]$.

What is the formula for a $2 \times 2$ determinant?

$$\det(A) = ad - bc$$

What is the determinant of a triangular matrix?

The product of its diagonal entries.

How does swapping two rows affect the determinant?

It multiplies the determinant by $-1$.

How does multiplying one row by $c$ affect the determinant?

It multiplies the determinant by $c$.

How does adding a multiple of one row to another affect the determinant?

It does not change the determinant.

What is the relationship between $\det(AB)$ and $\det(A)$ and $\det(B)$?

$$\det(AB) = \det(A) \det(B)$$

What is the relationship between $\det(A^T)$ and $\det(A)$?

$$\det(A^T) = \det(A)$$

What is $\det(A^{-1})$ for an invertible matrix $A$?

$$\det(A^{-1}) = \frac{1}{\det(A)}$$

What is a minor ($M_{ij}$)?

The determinant obtained after deleting row $i$ and column $j$.

What is a cofactor ($C_{ij}$)?

$$C_{ij} = (-1)^{i+j} M_{ij}$$

What is the geometric meaning of $|\det(A)|$ in $\mathbb{R}^2$?

The area scale factor of the transformation $\mathbf{x} \rightarrow \mathbf{Ax}$.

What does a negative determinant indicate regarding orientation?

It reverses orientation.

What is Cramer's Rule used for?

To solve a square system $\mathbf{Ax} = \mathbf{b}$ when $\det(A) \neq 0$.

What is the formula for the adjugate of $A$ ($\text{adj}(A)$)?

The transpose of the cofactor matrix.

What is the inverse formula using the adjugate?

$$A^{-1} = \frac{\text{adj}(A)}{\det(A)}$$

What are the three conditions for a subset to be a subspace?

1. Contains the zero vector; 2. Closed under vector addition; 3. Closed under scalar multiplication.

What is the null space of matrix $A$?

The set of solutions to $\mathbf{Ax} = \mathbf{0}$.

What is the span of a set of vectors?

The set of all possible linear combinations of those vectors.

How do you test if a vector $\mathbf{b}$ lies in $\text{Span}\{v_1, \dots, v_k\}$?

Solve the system $\begin{pmatrix} v_1 & v_2 & \dots & v_k \end{pmatrix} \mathbf{c} = \mathbf{b}$.

When are vectors $\mathbf{v}_1, \dots, \mathbf{v}_k$ linearly independent?

If $c_1 \mathbf{v}_1 + \dots + c_k \mathbf{v}_k = \mathbf{0}$ has only the trivial solution.

Can a set containing the zero vector be linearly independent?

No, any set containing the zero vector is linearly dependent.

If you have more than $n$ vectors in $\mathbb{R}^n$, what must be true?

The set of vectors must be linearly dependent.

What is a basis for a vector space?

A set of vectors that is both linearly independent and spans the vector space.

What is the dimension of a vector space?

The number of vectors in any basis for that space.

What are the standard basis vectors for $\mathbb{R}^2$?

$$\mathbf{e}_1 = \langle 1, 0 \rangle, \mathbf{e}_2 = \langle 0, 1 \rangle$$