Chapter 1 theorems

Question 1

Theorem 1

Answer 1

Each matrix is row equivalent to only one reduced echelon matrix.

Question 2

Theorem 2

Answer 2

A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column

Question 3

Theorem 3

Answer 3

If A is an m x n matrix, with columns a1,…, an, and if b is in Rm, then the matrix equation Ax = b has the same solution set as the vector equation x1a1 + x2a2+…xnan = b.

Question 4

Theorem 4

Answer 4

Let A be an m x n matrix. Then the following statements are logically equivalent.

a. For each b in Rm, the equation Ax=b has a solution
b. Each b in Rm is a linear combination of the columns of A
c. The columns of A span Rm
d, A has a pivot position in every row

Question 5

Theorem 5

Answer 5

If A is an m x n matrix, u and v are vectors in Rn, and c is a scalar, then
a. A(u+v)= Au + Av
b. A(cu)= c(Au)

Question 6

Theorem 7

Answer 6

An indexed set S = {v1,..,vp} of two or more vectors is linearly dependent if and only at least one of vectors in S is a linear cobination of the others

Question 7

Theorem 8

Answer 7

If a set contains more vectors than there are entries in each vector, then the set is linear dependent.

Question 8

Theorem 9

Answer 8

If a set S = {v1,..vp} in Rn contains the zero vector, then the set is linearly dependent

Question 9

Theorem 10

Answer 9

Let T: Rn -> Rm be a linear transformation. Then there exists a unique matrix A such that T(x) = Ax

Question 10

Theorem 11

Answer 10

Let T: Rn -> Rm be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution.

Question 11

Theorem 12

Answer 11

Let T: Rn -> Rm be a linear transformation and let A be the standard matrix for T. Then:

T maps Rn onto Rm if and only if the columns of A span Rm
b) T is one-to-one if and only if the columns are linearly independent.