Math 129 Chapters 3 and 4 (Test 2)

These three conditions yield a determinant of zero:

1. An entire row or column consists of zeros 2. Two rows or columns are equal 3. One row or column is a multiple of another row or column

List the five properties of vector multiplication

1. cu is a vector in the plane for a constant c and a vector u 2. c(u + v) = cu + cv 3. (c + d)u = cu + du 4. c(du) = (cd)u 5. 1(u) = u

List the 5 scalar multiplication axioms for vector spaces

1. cu is in the vector space, where u is a vector and c is a scalar 2. c(u + v) = cu + cv 3. (c + d)u = cu + du 4. c(du) = (cd)u 5. 1(u) = u

List the five properties of vector addition

1. u + v is a vector in the plane 2. u + v = v + u 3. (u+v)+w = u + (v+w) 4. u + 0 = u 5. u + (-u) = 0

List the 5 addition axioms for vector spaces

1. u + v is in the vector space 2. u + v = v + u 3. The be vector space V has a zero vector such that v + 0 = v for each vector in the vector space 4. u + (v + w) = (u + v) + w 5. For every vector v in the vector space, there is an element u such that v + u = 0. i.e. u = -v

Given a matrix A which is invertible and is n x n, then det(A^-1) =

1/det(A)

A vector space is composed of these four entities

A set of vectors, a set of scalars, vector addition, and scalar multiplication

Diagonal Matrix

A square matrix that is either upper triangular or lower traingular

What begins with T, ends with T, and has T in it?

A teapot

Subspace

Any vector space that is fully contained within a larger vector space

Additive Identity

For a vector v, the additive identity is the vector 0 v + 0 = v

Additive Inverse

For a vector v, the additive inverse is -v v + (-v) = 0

Proper subspace

Given a vector space V and a subspace of V called W, W is a proper subspace if and only if: 1. V is not equal to W 2. V is not equal to the zero vector {0}

Equal vectors

Given two vectors (x,y) and (a,b) , these vectors are only equal if x=a and y=b

Determinant of a matrix

If A is an nxn matrix, then its determinant is the sum of the entries in the first row of A multiplied by their respective cofactors

Intersection of vector spaces

If V and W are subspaces of X, then the intersection of V and W is also a subspace of X

Test for Subspace

If W is a nonempty subset of a vector space V, then W is a subspace of V if and only if: 1. u + v is in W for all vectors in W 2. If u is in W and c is a scalar, then cu is in W

Vector addition

If a = (r, s) and b = (h,g), then a + b = (r+h , s+g)

Standard operations in R (real numbers)

Summing two vectors and scalar multiplication

Initial point of a vector

The base of a vector located at the origin (0,0)

Terminal point of a vector

The tip of a vector. This location is denoted as (x,y), and indicates the direction of the vector

Ordered n-tuple

This represents a vector in n-space, denoted by (x1, x2, x3, ...... , xn) Example: a 5-tuple looks like (1, 6, -9, 4, 7)

Column equivalent

Two matrices are column equivalent if one can be obtained from the other by elementary column operations

If A is a square matrix of order n and c is a scalar, then the determinant of cA is det(cA) =

c^n * det(A)

If A is a square matrix, then det(A^T) [determinant of A transpose] is equal to

det(A)

Given to matrices A and B of order n, then det(AB) =

det(A)*det(B)

When finding the determinant through the use of row operations, adding a multiple of one row to another will ______

have no effect on the determinant

When finding the determinant through the use of row operations, switching to rows and columns will ___________

multiply the determinant by -1

When finding the determinant through the use of row operations, multiplying a row by a nonzero constant c will ______

multiply the resulting determinant by c

If a matrix is diagonal, you can then compute its determinant by

multiplying the entries along the main diagonal

A square matrix is invertible if and only if

the determinant of the matrix is not zero