Linear Algebra Definitions

Define orthogonal matrices.

A matrix M is orthogonal if its transpose MT equals the inverse M−1 of M.

What does it mean that N is the inverse of a matrix M?

A matrix N is the inverse of M if M⋅N equals the identity matrix.

Give a definition of a unit vector

A unit vector is a unit-length element of Euclidean space. Equivalently, one may say that the norm of a unit vector is equal to 1, and write ∥u⃗ ∥=1, where u⃗ is the vector in question.

Give a definition of a symmetric matrix M.

Definition: A square matrix M is called symmetric if M^T=M.

Give a definition of the kernel of a linear transformation of vector spaces. What is the connection of this definition to systems of linear equations?

Definition: If f:V→W is a linear transformation, then the kernel of f is {v∈V | f(v)=0}. If a system of homogeneous linear equations is converted to the matrix form, then it becomes M⋅X=0⃗ . The solution space of that system is identical to the kernel of the linear transformation f defined by f(X)=M⋅X and is called the null-space of M.

Give a definition of an orthogonal matrix M

Definition: M is orthogonal if its inverse and its transpose are equal (M^(−1) = M^t).

Give a definition of the matrix of a linear transformation f:Rn→Rm

Definition: M is the matrix of f:R^n→R^m if f(x)=M⋅x for all x∈R^n.

Give a definition of the rank of a matrix M.

Definition: The rank of M is the dimension of the space spanned by its columns (equivalently, rows).

Give a definition of the trace of a square matrix M.

Definition: The trace of a square matrix M is the sum of all elements "a_ii" on its diagonal.

Vector v is a linear combination of v1,...,vn. What does that mean?

Definition: Vector v is a linear combination of v1,...,vn if there are scalars c1,...,cn so that v=c1v1+...+cnvn.

Give a definition of the characteristic polynomial of a square matrix M

Definition: det(M−x⋅I) is a polynomial with the unknown x (usually denoted by λ). This polynomial is called the characteristic polynomial of the matrix M.

Give a geometrical (coordinate free) definition of the triple scalar product.

Definition: det[u⃗ ,v⃗ ,w⃗ ] is the number whose length equals the volume of the parallelepiped spanned by u⃗ , v⃗ and w⃗ and whose sign is determined by the right hand rule.

Give a definition of a linear transformation of vector spaces. What are two basic examples of linear transformations from calculus?

Definition: f:W→V is a linear transformation if V,W are vector spaces and f(a⋅v+b⋅w)=a⋅f(v)+b⋅f(w) for all a,b∈R and all v,w∈W. Basic linear transformations from calculus are the derivative and the definite integral.

Give a geometrical (coordinate free) definition of the double scalar product (determinant) for vectors on the plane.

Definition: u⃗ X_2 v⃗ is the number whose length equals the area of the parallelogram spanned by u⃗ and v⃗ and whose sign is determined by the right hand rule.

Give a geometrical (coordinate free) definition of the vector product.

Definition: u⃗ ×v⃗ is the unique vector perpendicular to both u⃗ and v⃗ whose length equals the area of the parallelogram spanned by u⃗ and v⃗ and whose direction is determined by the right hand rule.

Give a geometrical (coordinate free) definition of dot product.

Definition: u⃗ ⋅v⃗ is the product of their lengths and the cosine of the angle between them. If u and v are unit vectors, then their dot product u⋅v is the cosine of the angle between u and v. In particular, i⋅i=1, i⋅j=0, and so on. The special case\, u⋅u\, of scalar product is the scalar square of the vector u.\, In Rn it equals to the square of the length of u: u⋅u=∥u∥^2

Vectors v1,...,vn are linearly independent. What does that mean?

Definition: v1,...,vn are linearly independent iff c1v1+...+cnvn=0 implies c1=...=cn=0.

Define the transpose of a matrix.

Given a matrix M one can find its transpose MT according to the following rule: the (i,j) entry of MT equals the (j,i) entry of M.

Explain the parallelogram spanned by two geometric vectors. Give a parametrization of the parallelogram spanned by vectors u⃗ and v⃗ whose initial point is the origin.

Given two geometric vectors u⃗ and v⃗ , we can slide u⃗ along v⃗ and create the parallelogram spanned by u⃗ and v⃗ . Algebraically, it means that we pick the endpoint of the vector t⋅u⃗ , 0≤t≤1, and we add s⋅v⃗ for some 0 ≤ s ≤ 1. Thus the simplest parametrization of that parallelogram (in case the initial points of the vectors u⃗ and v⃗ is the origin) is t⋅u⃗ + s⋅v⃗ where 0 ≤ s, t ≤ 1.

Define the product of two matrices.

Given two matrices M and N one can find their product P=M⋅N according to the following rule: the (i,j) entry of P is the dot product of i-th row of M and j-th column of N. Thus, for the product to be defined, the number of columns of M must be equal to the number of rows of N. The simplest case is R⋅C, the product of a row vector R and a column vector C of the same number of entries.

Give a definition of the range of a linear transformation of vector spaces. What is the connection of it to systems of linear equations?

If f:V→W is a linear transformation, then the range of f is {w∈W | w=f(v) for some v∈V}. If a system of non-homogeneous linear equations is converted to the matrix form, then it becomes M⋅X=B. The space of vectors B for which there is a solution X is identical to the range of the linear transformation f defined by f(X)=M⋅X.

Give basic triple scalar products det(u,v,w), where u, v, and w range over basic vectors i, j, and k.

If two vectors are identical, the determinant is 0. Thus det(i,i,j)=0 and so on. det(i,j,k)=1 and flipping two vectors changes the sign.

Explain the parallelepiped spanned by three geometric vectors. Give a parametrization of that parralelepiped in case the initial points of the vectors u⃗ , v⃗ , and w⃗ is the origin.

In case of three geometric vectors we can slide the parallelogram spanned by u⃗ and v⃗ along w⃗ and create the parallelepiped spanned by u⃗ , v⃗ and w⃗ . Parallelepiped spanned by three vectors. Algebraically, it means that we pick the endpoint of the vector t⋅u⃗ , 0 ≤ t ≤ 1, and we add s⋅v⃗ +q⋅w⃗ for some 0 ≤ s ,q ≤ 1. Thus the simplest parametrization of that parallelepiped (in case the initial points of the vectors u⃗ , v⃗ , and w⃗ is the origin) is t⋅u⃗ +s⋅v⃗ +q⋅w⃗ where 0≤s ,t, q≤1.

Give a definition of a vector space.

In order for V to be a vector space, the following conditions must hold for all elements X,Y,Z in V and any scalars r,s in F: 1. Commutativity: X+Y=Y+X. 2. Associativity of vector addition: (X+Y)+Z=X+(Y+Z). 3. Additive identity: For all X, 0+X=X+0=X. 4. Existence of additive inverse: For any X, there exists a -X such that X+(-X)=0. 5. Associativity of scalar multiplication: r(sX)=(rs)X. 6. Distributivity of scalar sums: (r+s)X=rX+sX. 7. Distributivity of vector sums: r(X+Y)=rX+rY. 8. Scalar multiplication identity: 1X=X.

Give a geometrical interpretation of the determinant of a 3 by 3 matrix.

Interpretation: det[u⃗ ,v⃗ ,w⃗ ] is the number whose length equals the volume of the parallelepiped spanned by u⃗ , v⃗ and w⃗ and whose sign is determined by the right hand rule.

Give a geometrical interpretation of the determinant of a 2 by 2 matrix.

Interpretation: det[u⃗ ,v⃗ ] is the number whose length equals the area of the parallelogram spanned by u⃗ and v⃗ and whose sign is determined by the right hand rule.

What does it mean to normalize a non-zero vector v⃗ ? Give an example in the 3-space.

Let v⃗ be a non-zero vector. To normalize v⃗ is to find the unique unit vector with the same direction as v⃗ . This is done by multiplying v⃗ by the reciprocal of its length; the corresponding unit vector is given by u⃗ = v⃗ /∥v⃗ ∥. Example: Consider R^3 and the vector v⃗ =[1,2,3]. The norm (length) is Sqrt(14). Normalizing, we obtain the unit vector u⃗ pointing in the same direction, namely u⃗ =(1/√(14), 2/√(14), 3/√(14)).

Give an example of a non-euclidean vector space.

Minkowski spacetime (3 spatial dimension and one of time) This yields a vector space in which some element will not fulfill the standard requirements of a vector space.

Describe the Gram-Schmidt algorithm.

Suppose that we have a basis v1,...,vn of a Euclidean vector space V. The next procedure, called the Gram-Schmidt algorithm, produces an orthogonal basis w1,...,wn of V. Let w1=v1 The vector w2 appears in the parallel-perpendicular decomposition v2=x⋅v1+w2. Next, we can find w3 as v3−p3, where p3 is the orthogonal projection of v3 onto the plane spanned by w1 and w2. Continuing in this manner, we can get all vectors wi.

Given two vectors u⃗ and v⃗ what do we mean by the parallel-perpendicular decomposition of v⃗

That means expressing v⃗ as A⃗ +B⃗ , where A⃗ is parallel to u⃗ and B⃗ is perpendicular to u⃗ .

Explain how to find the area of the parallelogram spanned by non-zero vectors u⃗ and v⃗ using the angle between them.

The area of the parallelogram spanned by u⃗ and v⃗ is ∥u⃗ ∥⋅∥v⃗ ∥⋅sinφ where φ is the angle between u⃗ and v⃗ (recall, we consider the angle to be between 0 and 180 degrees).

Define the identity 3 by 3 matrix

The identity matrix In has its diagonal entries equal to 1 and off-diagonal entries are all 0.

State the form of the characteristic polynomial of a 2 by 2 matrix that uses the trace of M

Theorem: For a 2×2 matrix M, the characteristic polynomial is λ^2−Tr(M)λ+Det(M).

Define the vector component of a vector u⃗ with respect to vector v⃗ .

Think of v as pointing in the direction of the new x-axis. The new i-vector is inew = dir(v) = v/∥v∥ and the new j-vector, jnew, is on the plane spanned by v and u. The vector component of u in the direction of v is xnew⋅inew, the orthogonal projection of u onto v. Let θ be the angle from v to u. As in basic geometry, xnew=∥u∥cosθ and we can express it using the dot product: ∥u∥cosθ=∥inew∥⋅∥u∥cosθ=u⋅inew=u⋅dir(v=(u⋅v)/|v|, so the vector component of u in the direction of v is ((u⋅v)/|v|^2)⋅v.

Define the scalar component of a vector u⃗ with respect to vector v⃗

Think of v as pointing in the direction of the new x-axis. The new i-vector is inew=dir(v)=v/∥v∥ and the new j-vector, jnew, is on the plane spanned by v and u. The scalar component of u in the direction of v is xnew, the x-coordinate of the tip of u in the new coordinate system. Let θ be the angle from v to u. As in basic geometry, xnew=∥u∥cosθ and we can express it using the dot product: ∥u∥cosθ=∥inew∥⋅∥u∥cosθ=u⋅inew=u⋅dir(v=(u⋅v)/|v|.

Define orthogonal projection of a vector v⃗ onto vector u⃗ .

To project vector v⃗ orthogonally onto vector u⃗ means to find vector A⃗ parallel to u⃗ such that B⃗ :=u⃗ −A⃗ is perpendicular to u⃗ .

Define vectors geometrically

Vectors can be viewed as directed line segments (arrows) between two points in the plane (or in space), but with the stipulation that translating such segments (without turning) results in the same vector. Each vector v⃗ has its magnitude (also known as length or modulus) ∥v⃗ ∥ and its direction. Vectors can be added, and multiplied with (real) numbers. Numbers are called scalars when used in the context of vectors. Vectors are added as follows: To get a⃗ +b⃗ , move the tail of~b⃗ at the tip of~a⃗ ; then a⃗ +b⃗ goes from the tail of~a⃗ to the tip of~b⃗ , i.e., AB→+BC→=AC→. For k>0, k times a vector a⃗ , denoted ka⃗ has the same direction as a⃗ , but k times its length. For k<0, ka⃗ has the opposite direction as a⃗ , but |k| times its length.

Vectors v1,...,vn form a basis of a vector space V. What does that mean?

Vectors v1,...,vn form a basis of V iff they are linearly independent and all vectors of V are linear combinations of vectors v1,...,vn.

Explain the connection of the 3×3 determinant to cross product.

det(u,v,w)=(u×v)⋅w=u⋅(v×w)=v⋅(w×u).

Define algebraically the determinant of the matrix [[a11,a12],[a21,a22]]

det[[a11,a12],[a21,a22]] =a11⋅a22−a12⋅a21.

Derive (from the geometric definition) basic double scalar products u X v, where u and v range over basic vectors i and j

i X i=j X j=0 (the parallelogram has no area), i X j=1 (the parallelogram is the square of area 1 and vectors are positively oriented), j X i=−1 (the parallelogram is the square of area 1 and vectors are positively oriented).

Derive (from the geometric definition) basic cross products u×v, where u and v range over basic vectors i, j, and k.

i×i=j×j=k×k=0 (no area), i×j=k, j×i=−k, i×k=−j, k×i=j, j×k=i, k×j=−i.

Explain how to detect if non-zero vectors u⃗ and v⃗ are parallel using double scalar product.

u is parallel to v if only if the parallelogram spanned by them has area 0. Thus, the basic equation for two vectors being parallel is u X_2 v = 0.

Explain how to detect if non-zero vectors u⃗ and v⃗ are parallel using cross product.

u is parallel to v if only if the parallelogram spanned by them has area 0. Thus, the basic equation for two vectors being parallel is u×v=0⃗ .

Explain how to detect if non-zero vectors u⃗ and v⃗ are perpendicular (or orthogonal) using double scalar product.

u is parallel to v if only if the parallelogram spanned by them is a rectangle. A parallelogram spanned by u and v is a rectangle if and only if its area is the product of magnitudes of u and v. Thus, the basic equation for two vectors being perpendicular is |u X_2 v|=|u|⋅|v|.

Explain how to detect if non-zero vectors u⃗ and v⃗ are parallel using dot product.

u is parallel to v only if the angle α between them is 0 or 180 degrees. Also, |cos(α)|=1 if and only if α=0 or α=180 (we assume 0≤α≤180). Thus, the basic equation for two vectors being parallel is |cos(α)|=1. It is equivalent to |u|⋅|v|⋅|cos(α)|=|u|⋅|v| or |u⋅v|=|u|⋅|v|.

Explain how to detect if non-zero vectors u⃗ and v⃗ are perpendicular (or orthogonal) using cross product.

u is perpendicular to v if only if the parallelogram spanned by them is a rectangle. A parallelogram spanned by u and v is a rectangle if and only if its area is the product of magnitudes of u and v. Thus, the basic equation for two vectors being perpendicular is |u×v|=|u|⋅|v|.

Explain how to detect if non-zero vectors u⃗ , v⃗ , and w⃗ are coplanar using the triple scalar product.

u⃗ , v⃗ , and w⃗ are coplanar if only if the parallelepiped spanned by them has volume 0. Thus, the basic equation for three vectors being coplanar is det(u⃗ ,v⃗ ,w⃗ )=0.

Give a definition of an eigenvector of a matrix M.

v is an eigenvector of M if v≠0 and M⋅v=λ⋅v for some scalar λ.

Give a definition of an eigenvalue of a matrix M.

λ is an eigenvalue of M if M⋅v=λ⋅v for some vector v≠0.