Linear Algebra Exam 2 Theorems and Definitions (Chapters 3 & 4)

Definition of a determinant

For n≥2, the determinant of a square matrix A = [aij] is the sum of n terms of the form +/- a1j det Aij, with plus and minus signs alternating, where the entries a11, a12,...,a1n are from the first row of A. In symbols, det A = a11 det A11 - a12detA12 + ... + (-1)^(1+n)a1n det A1n.

Definition of cofactor expansion

If A = (aij) in a nxn matrix then the (ij)-cofactor of A is the number cij = (-1)^i+j det(Aij)

Theorem 1: finding the determinant

The det(A) can be calculated by doing a cofactor expansion across any row or any column

How can you find the det(A) if the matrix is not an upper triangular matrix without reducing it?

You can do a cofactor expansion across any row or column

Theorem 2: determinants of a triangular matrix

If A is a triangular matrix, then det(A) is the product of the entries on the main diagonal of A

If A is a triangular matrix, then how can you find det(A)?

Take the product on the main diagonal of A to find det(A)

Theorem 3: Properties of Determinants

Let A be a nxn square matrix

1. If a multiple of one row of A is added to another row to produce a matrix B, then det(B) = det(A)

2. If two rows of A are interchanged to promote B, then det(B) = -det(A)

3. If one row of A is multiplied by k to produce B, then det(B) = k * det(A)

Let A be a nxn square matrix. List the three properties of determinants that must apply

1. If a multiple of one row of A is added to another row to produce a matrix B, then det(B) = det(A)

2. If two rows of A are interchanged to promote B, then det(B) = -det(A)

3. If one row of A is multiplied by k to produce B, then det(B) = k * det(A)

When is a matrix A invertible? (or any matrix)

If and only if det(A) != 0

If A is an nxn matrix, what is det(A^T)?

det(A)

Theorem 6: Multiplicative property

If A and B are nxn matrices, then detAB = (detA)(detB)

If A and B are nxn matrices, and detAB = (detA)(detB), what is the property that explains this?

The multiplicative property

Suppose A can be reduced to echelon form U, using only row swaps and row replacement, then if Ais invertible then what is det(A) equal to?

It is equal to (-1)^s p where s = number of swaps and p = product of the pivot elements.

If A is invertible then what is det(A) equal to?

0

Theorem 7: Cramer's rule

Let A be an invertible nxn matrix. For any b in Rn, the unique solution x of Ax = b has entries given by

xi = [det(Ai(b))]/[det(A)], i = 1...n

Theorem 8: Inverse formula

Let A be an invertible nxm matrix. Then A^-1 = [1/det(A)]*adj(A)

If A is an 2x2 matrix, the area of the parallelogram determined by the columns of A is what? Similarly, if A is a 3x3 matrix, the volume of the parallelepiped determined by the columns of A is what?

|det(A)|

Definition of a vector space

A nonempty set V of objects, called vectors, on which are defined operations called addition and multiplication by scalars (real numbers) subject to ten axioms (rules). The axioms must hold for all vectors u,v, and w in V and for all scalars c and d

What are the ten axioms of a vector space?

1. The sum of u and v, denoted by u + v, is in V

2. u + v = v + u

3. (u + v) + w = u + (v + w)

4. There is a zero vector 0 in V such that u + 0 = u

5. For each u in V, there is a vector -u in V such that u+(-u)=0

6. The scalar multiple of u by c, denoted by cu is in V

7. c(u+v) = cu + cv

8. (c+d)u = cu + du

9. c(du) = (cd)u

10. 1u = u

What is a subspace?

A subset of a vector space that is itself a vector space under the same operations of addition and scalar multiplication

What are the three properties of a subspace corresponding to any set H in Rn?

1. The zero vector is in H.

2. For each u and v in H, the sum u + v is in H.

3. For each u in H and each scalar c, the vector cu is in H.

If v1... vp are in vector space V, then Span {v1 ... vp} is what to V?

A subspace of V

The _____ of a matrix A is the set of all solutions of the homogenous equation Ax = 0 in set notation

The null space

Definition of Column Space

The column space of an m x n matrix A, written as Col A, is the set of all linear combinations of the columns of A. If A= [a1 ... an]. Col A= Span {a1,....,an}

What is a column space of an mxn matrix A?

A subspace of Rm

A _______ T from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique vector T(x) such that:

1. T(u + v) = T(u) + T(v) for all u, v in V

2. T(cu) = cT(u) for all u in V and all scalars c

A linear transformation

What are the two rules of linear transformations that must be abided by?

1. T(u + v) = T(u) + T(v) for all u, v in V

2. T(cu) = cT(u) for all u in V and all scalars c

A _____ for a subspace H of Rn is a linearly independent set in H that spans H.

A basis

What are the two conditions for a basis? Let H be a subspace of vector space V where an indexed set of vectors B = {b1 ... bn} in V

1. B is a linearly independent set

2. The subspaces spanned by B coincides with Hi that is H = Span {b1 ... bn}

Theorem 5: The Spanning Set Theorem

Let S = {v1 ... vp} be a set in V and let H = Span {v1 ... vp} what are the two properties that must be upheld?

1. if one of the vectors in S, say vk, is a linear combination of the remaining vectors in S, then the set formed from S by removing vk still spans H

2. If H != {0}, some subset of S is a basis for H

What do the pivot columns of a matrix A form?

The basis for Col(A)

Theorem 7: Unique Representation Theorem

Let B = {b1 ... bn} be a basis for a vector space V. Then for each x in V, there exists a uniue set of scalars c1 ... cn such that:

x = c1b1 + ... + cnbn

Define what it means to be a isomorphic vector space

If T: V -> W is a one-to-one linear transformation, we say V and W are isomorphic vector spaces

What does it mean for a vector space to be isomorphic?

There exists a bijective linear transformation (an isomorphism) between the vectors that preserves the structure of vector addition and scalar multiplication

Theorem 9: Dimension of the vector space

If a vector space V has a basis B ={b1 ... bn} then any set in V containing more than n vectors must be linearly dependent

Theorem 10: If a vector space V has a basis of n vectors, then every basis of V must consist of how many vectors?

Must consist of n vectors

What does it mean for V to be finite-dimensional?

V is spanned by a finite set and the dimension is written as dim(V)

What does it mean for V to me infinite-dimensional?

V is not spanned by a finite set

The dimension of the zero vector space {0} is defined to be what?

Defined to be zero

Theorem 12: The Basis Theorem

Let V be a p-dimensional vector space, p >= 1. Any linearly independent set of exactly p elements in V is automatically a basis for V. Any set of exactly p elements that spans V is automatically a basis for V.

Theorem 13: Rank

If two matrices A and B are row equivalent, then their row spaces are the same. If B is in echelon form, the nonzero rows of B form a basis for the row space of A as well as for that of B

Definition of the row space

The set of all linear combinations of the row vectors

Definition of rank of A

The dimension of the column space of A