Math 2586 sample midterm 2

Hsian-Hua Tseng

October 2017

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Problems

1. Give the detailed deﬁnition of vector spaces.

2. Consider the following matrix:

A=

1−2 1

2−3 5

107

.

Find bases for the following subspaces:

(a) N(A), the null space of A.

(b) R(A), the range of A.

3. Consider the following vectors in R3:

v1= (3,5,0), v2= (6,4,2).

Find a vector u∈R3such that

(a) The length of uis 1.

(b) uis orthogonal to both v1and v2.

4. Determine whether the following matrix is invertible or not:

A=

−1−2 11

1 3 −15

0−1 5

.

If Ais invertible, compute A−1.

5. Determine the dimension of the row space of the following matrix:

A=

1−2−3

−4 5 −6

−9−8 7

10 −11 12

.

1

## Document Summary

Problems: give the detailed de nition of vector spaces, consider the following matrix: Find bases for the following subspaces: (a) n (a), the null space of a. (b) r(a), the range of a: consider the following vectors in r3: v1 = (3, 5, 0), v2 = (6, 4, 2). Find a vector u r3 such that (a) the length of u is 1. (b) u is orthogonal to both v1 and v2: determine whether the following matrix is invertible or not: 5: determine the dimension of the row space of the following matrix: