REAL LA FINAL

Solution of a System

A solution is a set of values for the variables that satisfies every equation in the system.

Consistent vs Inconsistent

A system is consistent if it has at least one solution; it is inconsistent if it has no solution.

Possible Numbers of Solutions

A linear system has either $0$ solutions, exactly $1$ solution, or infinitely many solutions.

Coefficient Matrix

The coefficient matrix records only the coefficients of the variables in a linear system.

Augmented Matrix

The augmented matrix $[A|b]$ is formed by attaching the constants column $b$ to the coefficient matrix $A$.

REF

REF means row echelon form: all nonzero rows are above zero rows, each leading entry is to the right of the one above it, and entries below each leading entry are zero.

RREF

RREF means reduced row echelon form: it is REF and each leading entry is $1$ and is the only nonzero entry in its column.

Elementary Row Operations

The three row operations are: swap two rows; multiply a row by a nonzero scalar; add a multiple of one row to another.

Equivalent Systems

Two linear systems are equivalent if they have the same solution set.

Pivot

A pivot is a leading entry in a row of an echelon form matrix.

Basic Variable

A basic variable corresponds to a pivot column.

Free Variable

A free variable corresponds to a non-pivot column and can take arbitrary parameter values.

General Solution

The general solution writes all variables in terms of the free variables, often in parametric vector form.

Homogeneous System

A homogeneous system has the form $Ax=0$ and always has at least the trivial solution $x=0$.

Parametric Vector Form

Write the solution as $x=p+s\,v_1+t\,v_2+\cdots$, where $p$ is a particular solution and the vectors multiply free parameters.

Matrix Size

If a matrix has $m$ rows and $n$ columns, its size is $m\times n$.

Matrix Addition

Matrix addition is defined only for matrices of the same size and is done entry-by-entry.

Scalar Multiplication

To multiply a matrix by a scalar, multiply every entry by that scalar.

Matrix Multiplication

If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is defined and has size $m\times p$.

Identity Matrix

The identity matrix $I_n$ is the $n\times n$ matrix with ones on the diagonal and zeros elsewhere.

Inverse of a Matrix

A square matrix $A$ is invertible if there exists a matrix $A^{-1}$ such that $AA^{-1}=A^{-1}A=I$.

How to Solve with an Inverse

If $A$ is invertible, the solution to $Ax=b$ is $x=A^{-1}b$.

Invertible Matrix Theorem — Core Ideas

For an $n\times n$ matrix $A$, these are equivalent: $A$ is invertible; $Ax=b$ has a unique solution for every $b$; $Ax=0$ has only the trivial solution; $A$ is row equivalent to $I_n$; $\det(A)\neq 0$; $\operatorname{rank}(A)=n$; the columns of $A$ span $\mathbb{R}^n$; the columns of $A$ are linearly independent.

Determinant of a $2\times 2$ Matrix

If $A=((a,b),(c,d))$, then $\det(A)=ad-bc$.

Meaning of Determinant

For a square matrix, $\det(A)\neq 0$ means the matrix is invertible; $\det(A)=0$ means it is not invertible.

Determinant Row Facts

Swapping two rows changes the sign of the determinant; multiplying a row by $c$ multiplies the determinant by $c$; adding a multiple of one row to another does not change the determinant.

Determinant of a Triangular Matrix

If $A$ is upper or lower triangular, then $\det(A)$ is the product of the diagonal entries.

Cofactor

The cofactor of entry $a_{ij}$ is $C_{ij}=(-1)^{i+j}M_{ij}$, where $M_{ij}$ is the minor obtained by deleting row $i$ and column $j$.

Adjoint Matrix

The adjoint matrix is $\operatorname{adj}(A)=C^T$, where $C$ is the cofactor matrix.

Inverse by Adjoint

If $\det(A)\neq 0$, then $A^{-1}=\dfrac{1}{\det(A)}\operatorname{adj}(A)$.

Cramer's Rule

If $A$ is invertible, the solution of $Ax=b$ satisfies $x_i=\dfrac{\det(A_i)}{\det(A)}$, where $A_i$ is obtained by replacing column $i$ of $A$ with $b$.

Vector in $\mathbb{R}^n$

A vector in $\mathbb{R}^n$ is an ordered $n$-tuple, written as $\langle x_1,x_2,\dots,x_n\rangle$.

Vector Addition

For vectors $u,v\in\mathbb{R}^n$, add componentwise: $u+v=\langle u_1+v_1,\dots,u_n+v_n\rangle$.

Scalar Multiple of a Vector

For scalar $c$ and vector $v$, $cv=\langle cv_1,\dots,cv_n\rangle$.

Norm of a Vector

The norm of $x=\left(x_1,\dots,x_n\right)$ is $|x|=\sqrt{x_1^2+\cdots+x_n^2}$.

Vector Space

A vector space is a set with vector addition and scalar multiplication satisfying the vector space axioms.

Subspace Test

A nonempty subset $S$ of a vector space is a subspace if it is closed under addition and scalar multiplication.

Fast Way to Show Not a Subspace

Show either $0\notin S$, or closure under addition fails, or closure under scalar multiplication fails.

Linear Combination

A vector $x$ is a linear combination of $v_1,\dots,v_k$ if $x=c_1v_1+\cdots+c_kv_k$ for some scalars.

Span

The span of ${v_1,\dots,v_k}$ is the set of all linear combinations of those vectors.

How to Test if $x\in \operatorname{Span}{v_1,\dots,v_k}$

Solve $c_1v_1+\cdots+c_kv_k=x$; if the system is consistent, then $x$ is in the span.

Linearly Independent

Vectors $v_1,\dots,v_k$ are linearly independent if $c_1v_1+\cdots+c_kv_k=0$ implies $c_1=\cdots=c_k=0$.

Linearly Dependent

Vectors are linearly dependent if there is a nontrivial solution to $c_1v_1+\cdots+c_kv_k=0$.

Quick Test for Two Vectors

Two vectors are linearly dependent iff one is a scalar multiple of the other.

Dependence Theorem

A set is linearly dependent iff at least one vector in the set can be written as a linear combination of the others.

Basis

A basis for a vector space is a set of vectors that is both linearly independent and spanning.

Standard Basis of $\mathbb{R}^n$

The standard basis is $e_1,e_2,\dots,e_n$, where each $e_i$ has a $1$ in position $i$ and zeros elsewhere.

Dimension

The dimension of a vector space is the number of vectors in any basis for that space.

Basis Shortcut in $\mathbb{R}^n$

If a set has exactly $n$ vectors in an $n$-dimensional space, then proving either independence or spanning is enough to conclude it is a basis.

Too Many Vectors Rule

Any set with more than $n$ vectors in an $n$-dimensional vector space is linearly dependent.

Row Space

The row space of a matrix is the span of its row vectors.

Column Space

The column space of a matrix is the span of its column vectors.

Rank of a Matrix

The rank of a matrix is $\dim(\operatorname{col}A)=\dim(\operatorname{row}A)$ and equals the number of pivots.

Basis for Row Space

A basis for the row space is given by the nonzero rows of an REF or RREF of the matrix.

Basis for Column Space

Find pivot columns in an REF or RREF, then take the corresponding original columns of $A$.

Null Space

The null space of $A$ is $\operatorname{Nul}(A)={x:Ax=0}$.

Nullity

The nullity of $A$ is $\dim(\operatorname{Nul}(A))$.

Rank-Nullity Theorem

If $A$ is $m\times n$, then $\operatorname{rank}(A)+\operatorname{nullity}(A)=n$.

Coordinate Vector

If $B={v_1,\dots,v_n}$ is an ordered basis and $x=c_1v_1+\cdots+c_nv_n$, then $[x]_B=\langle c_1,\dots,c_n\rangle$.

How to Find $[x]_B$

Solve $c_1v_1+\cdots+c_nv_n=x$ and use the coefficients as the coordinates.

Transition Matrix

If $P$ is the transition matrix from basis $B$ to basis $C$, then $P[x]_B=[x]_C$.

How to Find a Transition Matrix

Place the new basis and old basis into $[C\;B]$ and row reduce to $[I\;P]$ to get the transition matrix from $B$ to $C$.

Linear Transformation

A function $T:\mathbb{R}^n\to\mathbb{R}^m$ is linear if $T(u+v)=T(u)+T(v)$ and $T(cu)=cT(u)$.

Key Properties of Linear Transformations

If $T$ is linear, then $T(0)=0$, $T(-u)=-T(u)$, $T(u-v)=T(u)-T(v)$, and $T(cu+dv)=cT(u)+dT(v)$.

Matrix Transformation

Every $m\times n$ matrix $A$ defines a linear transformation $T(x)=Ax$ from $\mathbb{R}^n$ to $\mathbb{R}^m$.

Standard Matrix of $T$

The standard matrix of $T:\mathbb{R}^n\to\mathbb{R}^m$ is $A=[T(e_1)\;T(e_2)\;\cdots\;T(e_n)]$.

Composition of Linear Transformations

If $S$ has matrix $A_S$ and $T$ has matrix $A_T$, then $T\circ S$ has matrix $A_TA_S$.

Kernel of a Linear Transformation

The kernel is $\ker(T)={x:T(x)=0}$.

Image of a Linear Transformation

The image is $\operatorname{im}(T)={T(x):x\in\text{domain}}$.

Kernel and Null Space

If $A$ is the standard matrix of $T$, then $\ker(T)=\operatorname{Nul}(A)$.

Image and Column Space

If $A$ is the standard matrix of $T$, then $\operatorname{im}(T)=\operatorname{col}(A)$.

One-to-One Test

A linear transformation is one-to-one iff $\ker(T)={0}$.

Onto Test

A linear transformation $T:\mathbb{R}^n\to\mathbb{R}^m$ is onto iff $\operatorname{rank}(T)=m$.

Invertible Linear Transformation

A linear transformation $T:\mathbb{R}^n\to\mathbb{R}^n$ is invertible iff it is one-to-one iff it is onto iff its standard matrix is invertible.

Eigenvalue

A scalar $\lambda$ is an eigenvalue of $A$ if there exists a nonzero vector $x$ such that $Ax=\lambda x$.

Eigenvector

A nonzero vector $x$ satisfying $Ax=\lambda x$ is an eigenvector corresponding to eigenvalue $\lambda$.

Characteristic Equation

Eigenvalues are found from $\det(A-\lambda I)=0$.

How to Find Eigenvectors

For each eigenvalue $\lambda$, solve $(A-\lambda I)x=0$.

Eigenspace

The eigenspace for $\lambda$ is $E_\lambda={x:(A-\lambda I)x=0}$.

Triangular Matrix Eigenvalues

If $A$ is triangular, its eigenvalues are the diagonal entries.

Distinct Eigenvalues Rule

If an $n\times n$ matrix has $n$ distinct eigenvalues, then it has $n$ linearly independent eigenvectors.

Similar Matrices

Matrices $A$ and $B$ are similar if $B=P^{-1}AP$ for some invertible $P$.

Diagonalizable

A matrix $A$ is diagonalizable if it is similar to a diagonal matrix.

Diagonalization Test

An $n\times n$ matrix is diagonalizable iff it has $n$ linearly independent eigenvectors.

How to Build $P$ and $D$ for Diagonalization

Place linearly independent eigenvectors as the columns of $P$, and place the matching eigenvalues in the corresponding diagonal positions of $D$ so that $A=PDP^{-1}$.

Why Diagonalization Is Useful

If $A=PDP^{-1}$, then $A^k=PD^kP^{-1}$, and powers of diagonal matrices are easy because each diagonal entry is raised to the power $k$.

Complex Number Standard Form

A complex number has the form $a+bi$, where $a,b\in\mathbb{R}$ and $i^2=-1$.

Complex Conjugate

The conjugate of $z=a+bi$ is $\overline{z}=a-bi$.

Modulus of a Complex Number

The modulus is $|z|=\sqrt{a^2+b^2}$ for $z=a+bi$.

Division of Complex Numbers

To divide $\dfrac{z}{w}$, multiply numerator and denominator by the conjugate of the denominator.

Polar Form

A complex number can be written as $z=r(\cos\theta+i\sin\theta)$, where $r=|z|$.

Euler Form

The same polar form can be written as $z=re^{i\theta}$.

De Moivre's Theorem

If $z=r(\cos\theta+i\sin\theta)$, then $z^n=r^n(\cos(n\theta)+i\sin(n\theta))$.

Complex Eigenvalues

Some real matrices have complex eigenvalues, often found when the characteristic polynomial has no real roots.

Inner Product in $\mathbb{R}^n$

The standard inner product is $u\cdot v=u_1v_1+\cdots+u_nv_n$.

Orthogonal Vectors

Two vectors are orthogonal if $u\cdot v=0$.

Norm from Inner Product

The norm can be written as $|v|=\sqrt{v\cdot v}$.

Orthogonal Complement

If $W$ is a subspace of $\mathbb{R}^n$, then $W^\perp={x:x\cdot w=0\text{ for all }w\in W}$.

Orthogonal Set

A set of nonzero vectors is orthogonal if every pair of distinct vectors has dot product zero.

Orthogonal Set Independence

Any orthogonal set of nonzero vectors is linearly independent.