Exam 2

Question 1

Definition of a Linear Transformation

Answer 1

Question 2

Theorem 10 (1.8)

Answer 2

Question 3

onto vs one to one

Answer 3

A mapping T: Rⁿ -> R^m is said to be onto R^m if each b in R^m is the image of at least one x in Rⁿ

A mapping T: Rⁿ -> R^m is said to be one-to-one if each b in R^m is the image of at most one x in Rⁿ

Question 4

Theorem 11 (1.8)

Answer 4

Let T: Rⁿ -> R^m 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 5

Theorem 12 (1.8)

Answer 5

Let T: Rⁿ -> R^m be a linear transformation, and let A be the standard matrix for T.
Then:

• T maps Rⁿ onto R^m if and only if the columns of A span R^m
• T is one to one if and only if the columns of A are linearly independent

Question 6

Theorem 1 (2.1)

Answer 6

Question 7

Theorem 2 (2.1)

Answer 7

Question 8

Theorem 3 (2.1)

Answer 8

Question 9

Theorem 4 (2.2)

Answer 9

Question 10

Theorem 5 (2.2)

Answer 10

If A is a invertible n x n matri, then for each b in Rⁿ, the equation Ax=b has the unique solution x=A^-1b

Question 11

Theorem 6 (2.2)

Answer 11

If A is an invertible matrix then A^-1 is invertible and

• (A^-1)^-1 = A

If A and B are n x n invertible matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order. That is,

• (AB)^-1 = B^-1A^-1

If A is an invertible matrix, then so is A^T, and the inverse of A^T is the transpose of A^-1. Ths is,

• (A^T)^-1 = (A^-1)^T

Question 12

Theorem 7 (2.2)

Answer 12

An n x n matrix A is invertible if and only if A is row equivalent to I_n, and in this case, any sequence of elementary row operations that reduces A to I_n also transforms I_n to A^-1

Question 13

Theorem 8 IMT (2.3)

Answer 13

Let A be a square n x n matrix. Then the following statements are equivalent. For A, these are all either true or false:

• A is an invertible matrix
• A is row equivalent to the nxn identity matrix
• A has n pivot positions
• The equation Ax=0 has only the trivial solution
• The columns of A form a linearly independent set
• The linear transformation x |-> Ax maps Rⁿ onto Rⁿ
• There is an n x n matrix C such that CA = I
• There is an n x n matrix D such that AD = I
• A^T is an invertible matrix

Question 14

Theorem 9 (2.3)

Answer 14

Let T: Rⁿ -> Rⁿ be a linear transformation and let A be the standard matrix for T. Then T is invertible if and only if A is an invertible matrix. In that case, the linear transformation S given by S(x) = A^-1x is the unique function satisfying equations (1) and (2)

1. S(T(x)) = x for all x in Rⁿ
2. T(S(x)) = x for all x in Rⁿ

Question 15

Definition of a Vector Space

Answer 15

Let u,w,v be vectors in a vector space V

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 (additive inverse)
6. scalar multiples of a vector in V remains in V
7. c (u+v) = cu+cv
8. (c+d)v=cv+dv
9. c(du)=(cd)u
10. There exists a multiplication identity such that 1u=u

Question 16

Definition of a Subspace

Answer 16

A subspace of a vector space V is a subset H of V that has three properties:

• The zero vector of V is in H
• His closed under vector addition. That is, for each u and v in H, the sum u+v is in H
• H is closed under multiplication by scalars. That is, for each u in H and each scalar c, the vector cu is in H

Question 17

Theorem 1 (4.1)

Answer 17

If v₁,…,v_p are in a vector space V, then Span {v₁,…,v_p} is a subspace of V