哈工大选修课 LINEAR ALGEBRA 试卷及答案

LINEAR ALGEBRA

AND

ITS APPLICATIONS 姓名：易

学号：

成绩：

1. Definitions

(1) Pivot position in a matrix; (2) Echelon Form; (3) Elementary operations;

(4) Onto mapping and one-to-one mapping; (5) Linearly independence.

2. Describe the row reduction algorithm which produces a matrix in reduced echelon form.

3. Find the 33⨯ matrix that corresponds to the composite transformation of a scaling by 0.3, a rotation of 90︒, and finally a translation that adds (-0.5, 2) to each point of a figure.

4. Find a basis for the null space of the matrix

361171

22312

4

5

8

4A ---⎡⎤

⎢⎥=--⎢⎥⎢⎥--⎣⎦

5. Find a basis for Col A of the matrix

1

332-9-2-22-82230

7

13

4

-111

-8A ⎡⎤⎢⎥⎢

⎥=⎢⎥⎢⎥⎣⎦

6. Let a and b be positive numbers. Find the area of the region bounded by the ellipse whose equation is

222

2

1x y a

b

+

=

7. Provide twenty statements for the invertible matrix theorem. 8. Show and prove the Gram-Schmidt process. 9. Show and prove the diagonalization theorem.

10. Prove that the eigenvectors corresponding to distinct eigenvalues are linearly independent.

Answers:

1. Definitions

(1) Pivot position in a matrix:

A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position.

(2) Echelon Form:

A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties:

1.All nonzero rows are above any rows of all zeros.

2.Each leading entry of a row is in a column to the right of the leading entry of the row above it.

3.All entries in a column below a leading entry are zeros.

If a matrix in a echelon form satisfies the following additional conditions, then it is in reduced echelon form (or reduced row echelon form):

4.The leading entry in each nonzero row is 1.

5.Each leading 1 is the only nonzero entry in its column.

(3)Elementary operations:

Elementary operations can refer to elementary row operations or elementary column operations.

There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):

1.(Replacement) Replace one row by the sum of itself anda multiple of another row.

2.(Interchange) Interchange two rows.

3.(scaling) Multiply all entries in a row by a nonzero constant.

(4)Onto mapping and one-to-one mapping:

A mapping T : n →m is said to be onto m if each b in m is the image of at least one x in n.

A mapping T : n →m is said to be one-to-one if each b in m is the image of at most one x in n.

(5)Linearly independence:

An indexed set of vectors {V1, . . . ,V p} in n is said to be linearly independent if the vector equation